| L(s) = 1 | − 2-s + 4-s + 5-s − 3·7-s − 8-s − 3·9-s − 10-s − 3·13-s + 3·14-s + 16-s − 4·17-s + 3·18-s − 19-s + 20-s + 25-s + 3·26-s − 3·28-s − 3·29-s + 7·31-s − 32-s + 4·34-s − 3·35-s − 3·36-s − 8·37-s + 38-s − 40-s − 7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.832·13-s + 0.801·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.229·19-s + 0.223·20-s + 1/5·25-s + 0.588·26-s − 0.566·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.685·34-s − 0.507·35-s − 1/2·36-s − 1.31·37-s + 0.162·38-s − 0.158·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.51200092675394915806212248796, −9.726445409057165691739032513796, −9.024625262992485395383485370188, −8.146775621829067213517454637634, −6.84842964957324634340383682604, −6.25752170202416153036767439579, −5.05577133466160886295929369426, −3.31284984344814535566119778145, −2.25702097298873791784164571828, 0,
2.25702097298873791784164571828, 3.31284984344814535566119778145, 5.05577133466160886295929369426, 6.25752170202416153036767439579, 6.84842964957324634340383682604, 8.146775621829067213517454637634, 9.024625262992485395383485370188, 9.726445409057165691739032513796, 10.51200092675394915806212248796
