| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·7-s − 3·8-s + 9-s + 12-s − 13-s − 3·14-s − 16-s − 8·17-s + 18-s − 19-s + 3·21-s − 23-s + 3·24-s − 26-s − 27-s + 3·28-s + 5·29-s − 6·31-s + 5·32-s − 8·34-s − 36-s + 8·37-s − 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.13·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s − 0.801·14-s − 1/4·16-s − 1.94·17-s + 0.235·18-s − 0.229·19-s + 0.654·21-s − 0.208·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s + 0.566·28-s + 0.928·29-s − 1.07·31-s + 0.883·32-s − 1.37·34-s − 1/6·36-s + 1.31·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14434839267966, −12.84872029707258, −12.48761064922659, −12.01752659745317, −11.41737060980584, −11.04237033095707, −10.49580927545534, −9.934355716491138, −9.442350660212926, −9.176954346373835, −8.639881237366396, −8.032722375232991, −7.446501179713704, −6.639420897659785, −6.423227043146771, −6.190176183956925, −5.383645198475777, −4.947132233618483, −4.448406909837239, −4.032061954619700, −3.453906475307639, −2.832881258222365, −2.349771129560817, −1.497913306435227, −0.4741294205076146, 0,
0.4741294205076146, 1.497913306435227, 2.349771129560817, 2.832881258222365, 3.453906475307639, 4.032061954619700, 4.448406909837239, 4.947132233618483, 5.383645198475777, 6.190176183956925, 6.423227043146771, 6.639420897659785, 7.446501179713704, 8.032722375232991, 8.639881237366396, 9.176954346373835, 9.442350660212926, 9.934355716491138, 10.49580927545534, 11.04237033095707, 11.41737060980584, 12.01752659745317, 12.48761064922659, 12.84872029707258, 13.14434839267966
