| L(s) = 1 | − 2-s − 3·5-s + 4·7-s + 8-s + 3·10-s + 13-s − 4·14-s − 16-s + 6·17-s − 8·19-s + 5·25-s − 26-s + 9·29-s + 4·31-s − 6·34-s − 12·35-s − 2·37-s + 8·38-s − 3·40-s + 6·41-s − 8·43-s − 12·47-s + 7·49-s − 5·50-s + 12·53-s + 4·56-s − 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 1.51·7-s + 0.353·8-s + 0.948·10-s + 0.277·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 1.83·19-s + 25-s − 0.196·26-s + 1.67·29-s + 0.718·31-s − 1.02·34-s − 2.02·35-s − 0.328·37-s + 1.29·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 49-s − 0.707·50-s + 1.64·53-s + 0.534·56-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7541013370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7541013370\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−12.85359215630589938581548332257, −12.56092277731698895988134750242, −11.83773769543142733528718305468, −11.72820327580494862847660403966, −11.03558759433353539472538089512, −10.68085225261900373649094938870, −10.21551095504308587263727186398, −9.579885372707630361047891405271, −8.791583683002349525075838870308, −8.225980947945563300448616123922, −8.037100939293410167216438338055, −7.902141144098445941830875261832, −6.78184996924851581153455801714, −6.52059292934436182742534778741, −5.26256129327498560513843139936, −4.88409631804608462809178866932, −4.14601703291394264621390941545, −3.56603547037205579114074833461, −2.30857988801074308456157808772, −1.04188007843521944679127450189,
1.04188007843521944679127450189, 2.30857988801074308456157808772, 3.56603547037205579114074833461, 4.14601703291394264621390941545, 4.88409631804608462809178866932, 5.26256129327498560513843139936, 6.52059292934436182742534778741, 6.78184996924851581153455801714, 7.902141144098445941830875261832, 8.037100939293410167216438338055, 8.225980947945563300448616123922, 8.791583683002349525075838870308, 9.579885372707630361047891405271, 10.21551095504308587263727186398, 10.68085225261900373649094938870, 11.03558759433353539472538089512, 11.72820327580494862847660403966, 11.83773769543142733528718305468, 12.56092277731698895988134750242, 12.85359215630589938581548332257
