| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 4·7-s + 8-s − 10-s + 6·11-s + 13-s − 4·14-s − 15-s − 16-s − 3·17-s − 2·19-s − 4·21-s − 6·22-s − 24-s − 26-s + 27-s − 6·29-s + 30-s − 8·31-s − 6·33-s + 3·34-s + 4·35-s + 11·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.277·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.872·21-s − 1.27·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s − 1.04·33-s + 0.514·34-s + 0.676·35-s + 1.80·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576311695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576311695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_2$ | \( 1 - 11 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.878239054423569226636079936166, −9.633258673698703187860387889141, −9.065505156518732485211837234604, −8.999582932397830252761232864753, −8.508513689942342705592719183176, −8.051834098891151457851135003076, −7.56782466759933503910621492650, −7.29661070892775548685309680603, −6.65102638732774285269019993368, −6.21966732912253373981346406212, −6.00643948418209814195783956982, −5.27649957827327823292054789467, −4.99890206292190340404723554584, −4.26007425008852788319433041974, −4.12805427966953877989478370948, −3.51204786983622105353654680162, −2.46758968551662027939843231709, −1.84743324634711795347959393023, −1.49088098446221964131153324016, −0.69945685765039033126907425969,
0.69945685765039033126907425969, 1.49088098446221964131153324016, 1.84743324634711795347959393023, 2.46758968551662027939843231709, 3.51204786983622105353654680162, 4.12805427966953877989478370948, 4.26007425008852788319433041974, 4.99890206292190340404723554584, 5.27649957827327823292054789467, 6.00643948418209814195783956982, 6.21966732912253373981346406212, 6.65102638732774285269019993368, 7.29661070892775548685309680603, 7.56782466759933503910621492650, 8.051834098891151457851135003076, 8.508513689942342705592719183176, 8.999582932397830252761232864753, 9.065505156518732485211837234604, 9.633258673698703187860387889141, 9.878239054423569226636079936166
