| L(s) = 1 | + 3-s − 2·4-s − 5-s − 2·7-s + 9-s − 4·11-s − 2·12-s − 3·13-s − 15-s + 4·16-s − 4·19-s + 2·20-s − 2·21-s − 3·23-s − 4·25-s + 27-s + 4·28-s + 5·29-s − 5·31-s − 4·33-s + 2·35-s − 2·36-s + 5·37-s − 3·39-s + 4·41-s − 4·43-s + 8·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.832·13-s − 0.258·15-s + 16-s − 0.917·19-s + 0.447·20-s − 0.436·21-s − 0.625·23-s − 4/5·25-s + 0.192·27-s + 0.755·28-s + 0.928·29-s − 0.898·31-s − 0.696·33-s + 0.338·35-s − 1/3·36-s + 0.821·37-s − 0.480·39-s + 0.624·41-s − 0.609·43-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−10.57493263175513756189625933024, −9.910142982158485475927583257856, −9.065575868257068825573288642415, −8.113522334387238488472324316332, −7.44353068278242072414656558921, −5.98085143559561728984920629207, −4.76397447600939130338286205555, −3.80807966629184542405043254266, −2.57337133630687979423683709536, 0,
2.57337133630687979423683709536, 3.80807966629184542405043254266, 4.76397447600939130338286205555, 5.98085143559561728984920629207, 7.44353068278242072414656558921, 8.113522334387238488472324316332, 9.065575868257068825573288642415, 9.910142982158485475927583257856, 10.57493263175513756189625933024
