| L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s + 4·13-s + 7·17-s − 3·19-s − 9·21-s − 9·27-s + 7·29-s + 3·31-s − 9·37-s − 12·39-s + 4·41-s + 6·43-s + 12·47-s + 2·49-s − 21·51-s + 7·53-s + 9·57-s − 12·59-s + 61-s + 18·63-s − 12·67-s + 9·71-s + 2·73-s + 12·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s + 1.10·13-s + 1.69·17-s − 0.688·19-s − 1.96·21-s − 1.73·27-s + 1.29·29-s + 0.538·31-s − 1.47·37-s − 1.92·39-s + 0.624·41-s + 0.914·43-s + 1.75·47-s + 2/7·49-s − 2.94·51-s + 0.961·53-s + 1.19·57-s − 1.56·59-s + 0.128·61-s + 2.26·63-s − 1.46·67-s + 1.06·71-s + 0.234·73-s + 1.35·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.428408061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.428408061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.84689981533478, −12.31061771413937, −12.17120541474693, −11.76889059244124, −11.10108437838900, −10.84915283144951, −10.42033828131399, −10.15263592430401, −9.373102346856813, −8.741352603844400, −8.311259181298565, −7.705320959383274, −7.334131550413373, −6.691397947905699, −6.069499371767832, −5.904007716927079, −5.297087326369544, −4.853511327077099, −4.391394488192658, −3.837640529240773, −3.184685513435542, −2.257103403826083, −1.535222155743313, −1.004609612043902, −0.6300461838284679,
0.6300461838284679, 1.004609612043902, 1.535222155743313, 2.257103403826083, 3.184685513435542, 3.837640529240773, 4.391394488192658, 4.853511327077099, 5.297087326369544, 5.904007716927079, 6.069499371767832, 6.691397947905699, 7.334131550413373, 7.705320959383274, 8.311259181298565, 8.741352603844400, 9.373102346856813, 10.15263592430401, 10.42033828131399, 10.84915283144951, 11.10108437838900, 11.76889059244124, 12.17120541474693, 12.31061771413937, 12.84689981533478
