| L(s) = 1 | + 3-s − 2·4-s + 2·5-s + 2·7-s + 9-s − 5·11-s − 2·12-s + 3·13-s + 2·15-s + 4·16-s − 3·17-s − 2·19-s − 4·20-s + 2·21-s − 23-s − 25-s + 27-s − 4·28-s − 5·31-s − 5·33-s + 4·35-s − 2·36-s − 8·37-s + 3·39-s − 7·41-s + 10·44-s + 2·45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s + 0.832·13-s + 0.516·15-s + 16-s − 0.727·17-s − 0.458·19-s − 0.894·20-s + 0.436·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.898·31-s − 0.870·33-s + 0.676·35-s − 1/3·36-s − 1.31·37-s + 0.480·39-s − 1.09·41-s + 1.50·44-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.059592290888346365420813482711, −7.24650891152641622311326767857, −6.25092156557851862967201238599, −5.38353481546129357418197442687, −5.02270173768800302474938329281, −4.10857267207543274426489530992, −3.29981924554613740035491051437, −2.22633750088123189512744210341, −1.54997112185557038907193599436, 0,
1.54997112185557038907193599436, 2.22633750088123189512744210341, 3.29981924554613740035491051437, 4.10857267207543274426489530992, 5.02270173768800302474938329281, 5.38353481546129357418197442687, 6.25092156557851862967201238599, 7.24650891152641622311326767857, 8.059592290888346365420813482711
