| L(s) = 1 | − 3-s − 5-s − 5·7-s − 2·9-s + 2·11-s − 13-s + 15-s − 3·17-s + 2·19-s + 5·21-s − 4·23-s − 4·25-s + 5·27-s − 6·29-s + 4·31-s − 2·33-s + 5·35-s + 11·37-s + 39-s + 8·41-s + 43-s + 2·45-s − 9·47-s + 18·49-s + 3·51-s − 12·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 1.09·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s + 0.845·35-s + 1.80·37-s + 0.160·39-s + 1.24·41-s + 0.152·43-s + 0.298·45-s − 1.31·47-s + 18/7·49-s + 0.420·51-s − 1.64·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.89074471324911579422736349153, −11.11247624803048321318765362970, −9.822337358396720600057739846210, −9.192968041845742079013702027529, −7.74969917443740327809424196061, −6.47800053392196071138964912551, −5.90495037317248100686416791938, −4.17978609463531179755283405261, −2.93919437468071375351517854482, 0,
2.93919437468071375351517854482, 4.17978609463531179755283405261, 5.90495037317248100686416791938, 6.47800053392196071138964912551, 7.74969917443740327809424196061, 9.192968041845742079013702027529, 9.822337358396720600057739846210, 11.11247624803048321318765362970, 11.89074471324911579422736349153
