| L(s) = 1 | + 32·5-s + 32·9-s + 308·25-s + 624·29-s − 192·41-s + 1.02e3·45-s + 160·49-s + 2.11e3·61-s − 764·81-s + 80·89-s − 3.53e3·101-s + 3.39e3·109-s − 664·121-s − 1.69e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.99e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.41e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.86·5-s + 1.18·9-s + 2.46·25-s + 3.99·29-s − 0.731·41-s + 3.39·45-s + 0.466·49-s + 4.43·61-s − 1.04·81-s + 0.0952·89-s − 3.48·101-s + 2.98·109-s − 0.498·121-s − 1.21·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 11.4·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.46·169-s + 0.000439·173-s + 0.000417·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(23.62666310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.62666310\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 16 T + 46 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 16 T^{2} + 766 T^{4} - 16 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 80 T^{2} + 212622 T^{4} - 80 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 + 332 T^{2} + 474102 T^{4} + 332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2708 T^{2} + 7852758 T^{4} - 2708 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4450 T^{2} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2644 T^{2} + 87237846 T^{4} - 2644 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 36816 T^{2} + 602323406 T^{4} - 36816 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 156 T + 33358 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 95612 T^{2} + 4010875782 T^{4} + 95612 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 160436 T^{2} + 8226342 p^{2} T^{4} - 160436 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 48 T + 124222 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 66320 T^{2} - 2039763618 T^{4} - 66320 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 204240 T^{2} + 24337141742 T^{4} - 204240 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 444020 T^{2} + 93573616758 T^{4} - 444020 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 + 507788 T^{2} + 128506595382 T^{4} + 507788 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 528 T + 462422 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 - 814480 T^{2} + 336078840702 T^{4} - 814480 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 1385052 T^{2} + 735523082342 T^{4} + 1385052 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 971620 T^{2} + 454375527078 T^{4} - 971620 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 + 756668 T^{2} + 456844629702 T^{4} + 756668 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 300176 T^{2} + 429308372286 T^{4} - 300176 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 20 T + 872438 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 534850 T^{2} + p^{6} T^{4} )^{4} \) |
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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
−5.30427269549830813412618538750, −5.29429227994771338051606049269, −5.23731802382952637500698176788, −4.91649478382524877285923176531, −4.68967609455711988711723475681, −4.64008319903868148717623633521, −4.38488680841972823627493930141, −4.35992245518714077008759898351, −3.92580112744876274726015212916, −3.79901246545787289774327788112, −3.76621696162690482758448157665, −3.55167686017964275347178401325, −3.08165116074695062012308450986, −2.84104206802738002734833689816, −2.81761714331742832738234726732, −2.54259038736404796202911993060, −2.48600369849055589303356204426, −2.11307239092703246920526119746, −1.86548085766597083425000003769, −1.72246242498007187356295720300, −1.56193145737809749519729641611, −1.23966192492169492694470605436, −0.847848186583845727473206985292, −0.74177337154159347003225054451, −0.40005941588809582139455162648,
0.40005941588809582139455162648, 0.74177337154159347003225054451, 0.847848186583845727473206985292, 1.23966192492169492694470605436, 1.56193145737809749519729641611, 1.72246242498007187356295720300, 1.86548085766597083425000003769, 2.11307239092703246920526119746, 2.48600369849055589303356204426, 2.54259038736404796202911993060, 2.81761714331742832738234726732, 2.84104206802738002734833689816, 3.08165116074695062012308450986, 3.55167686017964275347178401325, 3.76621696162690482758448157665, 3.79901246545787289774327788112, 3.92580112744876274726015212916, 4.35992245518714077008759898351, 4.38488680841972823627493930141, 4.64008319903868148717623633521, 4.68967609455711988711723475681, 4.91649478382524877285923176531, 5.23731802382952637500698176788, 5.29429227994771338051606049269, 5.30427269549830813412618538750
Plot not available for L-functions of degree greater than 10.
