| L(s) = 1 | − 20·9-s + 8·11-s − 24·23-s − 20·25-s + 208·29-s − 40·37-s + 368·43-s + 152·53-s + 8·67-s + 496·71-s − 248·79-s + 230·81-s − 160·99-s + 328·107-s + 88·109-s + 336·113-s + 252·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 424·169-s + ⋯ |
| L(s) = 1 | − 2.22·9-s + 8/11·11-s − 1.04·23-s − 4/5·25-s + 7.17·29-s − 1.08·37-s + 8.55·43-s + 2.86·53-s + 8/67·67-s + 6.98·71-s − 3.13·79-s + 2.83·81-s − 1.61·99-s + 3.06·107-s + 0.807·109-s + 2.97·113-s + 2.08·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.50·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(25.20803881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.20803881\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 20 T^{2} + 170 T^{4} + 1360 T^{6} + 13939 T^{8} + 1360 p^{4} T^{10} + 170 p^{8} T^{12} + 20 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 + 4 p T^{2} + 618 T^{4} - 5872 p T^{6} - 593101 T^{8} - 5872 p^{5} T^{10} + 618 p^{8} T^{12} + 4 p^{13} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 4 T - 102 T^{2} + 496 T^{3} - 2653 T^{4} + 496 p^{2} T^{5} - 102 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 212 T^{2} + 14566 T^{4} - 212 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 + 900 T^{2} + 448650 T^{4} + 174877200 T^{6} + 55946055059 T^{8} + 174877200 p^{4} T^{10} + 448650 p^{8} T^{12} + 900 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 + 404 T^{2} - 121302 T^{4} + 9645904 T^{6} + 42696456947 T^{8} + 9645904 p^{4} T^{10} - 121302 p^{8} T^{12} + 404 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 + 12 T - 438 T^{2} - 5712 T^{3} - 17293 T^{4} - 5712 p^{2} T^{5} - 438 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 52 T + 2230 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( 1 + 3332 T^{2} + 6512394 T^{4} + 9138969616 T^{6} + 9960511380755 T^{8} + 9138969616 p^{4} T^{10} + 6512394 p^{8} T^{12} + 3332 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 20 T - 1286 T^{2} - 21040 T^{3} + 420835 T^{4} - 21040 p^{2} T^{5} - 1286 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 1284 T^{2} + 1246278 T^{4} - 1284 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 46 T + p^{2} T^{2} )^{8} \) |
| 47 | \( 1 + 1412 T^{2} + 5506698 T^{4} - 18740510192 T^{6} - 21199275571309 T^{8} - 18740510192 p^{4} T^{10} + 5506698 p^{8} T^{12} + 1412 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 76 T - 774 T^{2} - 70832 T^{3} + 18787235 T^{4} - 70832 p^{2} T^{5} - 774 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 10324 T^{2} + 58194922 T^{4} + 249379647568 T^{6} + 903238257150259 T^{8} + 249379647568 p^{4} T^{10} + 58194922 p^{8} T^{12} + 10324 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( 1 + 7252 T^{2} + 25564714 T^{4} - 4821796784 T^{6} - 189433537628045 T^{8} - 4821796784 p^{4} T^{10} + 25564714 p^{8} T^{12} + 7252 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 4 T - 8454 T^{2} + 2032 T^{3} + 51517955 T^{4} + 2032 p^{2} T^{5} - 8454 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 124 T + 13638 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 73 | \( 1 + 18180 T^{2} + 192317130 T^{4} + 1479829965840 T^{6} + 8905501769790419 T^{8} + 1479829965840 p^{4} T^{10} + 192317130 p^{8} T^{12} + 18180 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( ( 1 + 124 T - 150 T^{2} + 377456 T^{3} + 106144979 T^{4} + 377456 p^{2} T^{5} - 150 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 16404 T^{2} + 162018918 T^{4} - 16404 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 + 25476 T^{2} + 370895178 T^{4} + 3888832832016 T^{6} + 32699103900935507 T^{8} + 3888832832016 p^{4} T^{10} + 370895178 p^{8} T^{12} + 25476 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 4356 T^{2} + 109417734 T^{4} - 4356 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.36714836705507002183229263033, −4.04332430712252488749810298671, −4.03506421079454222600380082852, −3.64976085696301765906779925109, −3.64725146546249918614892289639, −3.60887385625852891053471051909, −3.36833344735427348903426962573, −3.29780506351609355276632370947, −3.16140785761199326391264998857, −2.72307682395979019658573714758, −2.66372436834614639178056192471, −2.62624909585047413325785137234, −2.55332649753593173583682657237, −2.41218953172449749215895736932, −2.25623576531125282270786930459, −2.24151522663370733203157413592, −1.97083532282114661805479504145, −1.60555153475043025541025551525, −1.42222481735802748727631061002, −1.03570662777058632676433880129, −0.836942604820889163097527299409, −0.74890658811755181233828003111, −0.70539796987481426467742208525, −0.64928672881310073065803815586, −0.37468223798587357430620510646,
0.37468223798587357430620510646, 0.64928672881310073065803815586, 0.70539796987481426467742208525, 0.74890658811755181233828003111, 0.836942604820889163097527299409, 1.03570662777058632676433880129, 1.42222481735802748727631061002, 1.60555153475043025541025551525, 1.97083532282114661805479504145, 2.24151522663370733203157413592, 2.25623576531125282270786930459, 2.41218953172449749215895736932, 2.55332649753593173583682657237, 2.62624909585047413325785137234, 2.66372436834614639178056192471, 2.72307682395979019658573714758, 3.16140785761199326391264998857, 3.29780506351609355276632370947, 3.36833344735427348903426962573, 3.60887385625852891053471051909, 3.64725146546249918614892289639, 3.64976085696301765906779925109, 4.03506421079454222600380082852, 4.04332430712252488749810298671, 4.36714836705507002183229263033
Plot not available for L-functions of degree greater than 10.
