| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s + 2·11-s + 4·13-s − 14-s + 16-s − 2·17-s + 3·18-s + 4·19-s − 20-s − 2·22-s + 25-s − 4·26-s + 28-s − 10·29-s − 2·31-s − 32-s + 2·34-s − 35-s − 3·36-s + 10·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.917·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.11657296954018, −14.68074875703946, −14.24946833774875, −13.55428501526709, −13.12808875442268, −12.32663799029861, −11.81522693123492, −11.28758887049981, −11.01501894159978, −10.63133039962324, −9.508338064514785, −9.272775490321166, −8.808957554714013, −8.160009869815569, −7.672059530425788, −7.231484334666903, −6.399880659969619, −5.824584088438799, −5.503827403101348, −4.401148789662664, −3.925604673295522, −3.172782362945885, −2.557511855340278, −1.624093436946765, −0.9678984612875988, 0,
0.9678984612875988, 1.624093436946765, 2.557511855340278, 3.172782362945885, 3.925604673295522, 4.401148789662664, 5.503827403101348, 5.824584088438799, 6.399880659969619, 7.231484334666903, 7.672059530425788, 8.160009869815569, 8.808957554714013, 9.272775490321166, 9.508338064514785, 10.63133039962324, 11.01501894159978, 11.28758887049981, 11.81522693123492, 12.32663799029861, 13.12808875442268, 13.55428501526709, 14.24946833774875, 14.68074875703946, 15.11657296954018
