| L(s) = 1 | − 40·23-s − 4·49-s + 56·53-s + 32·79-s − 72·107-s + 16·109-s − 24·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8.34·23-s − 4/7·49-s + 7.69·53-s + 3.60·79-s − 6.96·107-s + 1.53·109-s − 2.25·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 6.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613413821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613413821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 22 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 + 4 T^{2} - 10 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 40 T^{2} + 710 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 56 T^{2} + 1334 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 24 T^{2} + 838 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10 T + 64 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 48 T^{2} + 1126 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 116 T^{2} + 5990 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 24 T^{2} + 2134 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 44 T^{2} + 1382 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 84 T^{2} + 6070 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 14 T + 148 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 84 T^{2} + 3238 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 92 T^{2} + 3926 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 216 T^{2} + 20950 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 164 T^{2} + 15014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 336 T^{2} + 44038 T^{4} + 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 24 T^{2} + 17590 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.10359507634935059007320099028, −4.07172993921936618547218890159, −3.98388970645222604395035231386, −3.77814742756227019016096287374, −3.76449862467358939965175734071, −3.58541518839028186980065036630, −3.46169361443652963394168300138, −3.22015868453474389991826070062, −3.18984076459147962456233914900, −2.81775087891245844873887001937, −2.69161651052109277193684847416, −2.55217136621822151663093522581, −2.36900621091324165336763059437, −2.29089283982403971794506196687, −2.20896245567547400786008298527, −2.19669516295470184074007328987, −1.98298992952777797434967355129, −1.81657778943203841402390537436, −1.48476288833890131988092536077, −1.41915419380110760244835623013, −1.35598878540909519663169274307, −0.853653300453039662919897468046, −0.58784862225723321182678056790, −0.46608902697854590629902737597, −0.18718898160629468315075831989,
0.18718898160629468315075831989, 0.46608902697854590629902737597, 0.58784862225723321182678056790, 0.853653300453039662919897468046, 1.35598878540909519663169274307, 1.41915419380110760244835623013, 1.48476288833890131988092536077, 1.81657778943203841402390537436, 1.98298992952777797434967355129, 2.19669516295470184074007328987, 2.20896245567547400786008298527, 2.29089283982403971794506196687, 2.36900621091324165336763059437, 2.55217136621822151663093522581, 2.69161651052109277193684847416, 2.81775087891245844873887001937, 3.18984076459147962456233914900, 3.22015868453474389991826070062, 3.46169361443652963394168300138, 3.58541518839028186980065036630, 3.76449862467358939965175734071, 3.77814742756227019016096287374, 3.98388970645222604395035231386, 4.07172993921936618547218890159, 4.10359507634935059007320099028
Plot not available for L-functions of degree greater than 10.
