| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 11-s − 12-s − 4·13-s + 2·15-s + 16-s − 17-s + 18-s − 2·19-s − 2·20-s − 22-s + 2·23-s − 24-s − 25-s − 4·26-s − 27-s + 2·29-s + 2·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76197354834221, −14.20352843668462, −13.58139204633506, −13.01203697570255, −12.63722272281489, −11.99309488720483, −11.83400402612577, −11.14520080656308, −10.82969117717415, −10.06452967287679, −9.757073523372817, −8.919632103619421, −8.232971238957922, −7.719462434006789, −7.311033956050295, −6.623900568537516, −6.245742666510649, −5.469811682357478, −4.865677705183918, −4.540685571567346, −3.930477798277575, −3.212800397248715, −2.571736468419658, −1.875591991567310, −0.8135538415647749, 0,
0.8135538415647749, 1.875591991567310, 2.571736468419658, 3.212800397248715, 3.930477798277575, 4.540685571567346, 4.865677705183918, 5.469811682357478, 6.245742666510649, 6.623900568537516, 7.311033956050295, 7.719462434006789, 8.232971238957922, 8.919632103619421, 9.757073523372817, 10.06452967287679, 10.82969117717415, 11.14520080656308, 11.83400402612577, 11.99309488720483, 12.63722272281489, 13.01203697570255, 13.58139204633506, 14.20352843668462, 14.76197354834221
