| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·15-s + 4·17-s − 4·19-s + 4·23-s − 25-s + 27-s − 2·29-s + 4·31-s + 4·33-s − 12·37-s − 12·41-s + 8·43-s − 2·45-s + 4·51-s + 14·53-s − 8·55-s − 4·57-s + 2·59-s + 2·61-s + 4·67-s + 4·69-s − 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s − 1.97·37-s − 1.87·41-s + 1.21·43-s − 0.298·45-s + 0.560·51-s + 1.92·53-s − 1.07·55-s − 0.529·57-s + 0.260·59-s + 0.256·61-s + 0.488·67-s + 0.481·69-s − 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.42218960452151, −12.23848909676194, −11.87315506883407, −11.43872744101377, −10.85878414410327, −10.37902859256366, −10.02205893609218, −9.423290363623389, −8.929310429380651, −8.584147472110532, −8.247675078487645, −7.658291368395070, −7.149221974433718, −6.874131323743906, −6.349609288080453, −5.706296615584300, −5.148008309714651, −4.637843052013681, −3.963786155426466, −3.722207525706876, −3.344977195994111, −2.630901394612350, −2.001229082702602, −1.391858932223985, −0.8207135156714110, 0,
0.8207135156714110, 1.391858932223985, 2.001229082702602, 2.630901394612350, 3.344977195994111, 3.722207525706876, 3.963786155426466, 4.637843052013681, 5.148008309714651, 5.706296615584300, 6.349609288080453, 6.874131323743906, 7.149221974433718, 7.658291368395070, 8.247675078487645, 8.584147472110532, 8.929310429380651, 9.423290363623389, 10.02205893609218, 10.37902859256366, 10.85878414410327, 11.43872744101377, 11.87315506883407, 12.23848909676194, 12.42218960452151
