| L(s) = 1 | − 3-s − 2·4-s − 2·5-s + 3·7-s + 9-s − 11-s + 2·12-s + 2·15-s + 4·16-s + 2·17-s − 4·19-s + 4·20-s − 3·21-s − 4·23-s − 25-s − 27-s − 6·28-s − 5·31-s + 33-s − 6·35-s − 2·36-s + 6·37-s + 8·41-s + 5·43-s + 2·44-s − 2·45-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s + 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 0.516·15-s + 16-s + 0.485·17-s − 0.917·19-s + 0.894·20-s − 0.654·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.13·28-s − 0.898·31-s + 0.174·33-s − 1.01·35-s − 1/3·36-s + 0.986·37-s + 1.24·41-s + 0.762·43-s + 0.301·44-s − 0.298·45-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.63459838591067509196137494048, −7.53852067272613935468723009775, −6.10508693146509044491452013284, −5.58361392652687152697389899980, −4.70647816727076936936892026019, −4.25992637892603164738788019738, −3.62810083486290031377554077628, −2.24186505489398896412566670582, −1.05039285209905061302618939969, 0,
1.05039285209905061302618939969, 2.24186505489398896412566670582, 3.62810083486290031377554077628, 4.25992637892603164738788019738, 4.70647816727076936936892026019, 5.58361392652687152697389899980, 6.10508693146509044491452013284, 7.53852067272613935468723009775, 7.63459838591067509196137494048
