| L(s) = 1 | − 2.77·5-s − 2.49·7-s + 4.39·11-s − 4.76·13-s − 2.61·17-s + 4.10·19-s − 1.98·23-s + 2.70·25-s − 4.38·29-s − 2.65·31-s + 6.92·35-s + 37-s + 5.61·41-s − 11.1·43-s + 10.6·47-s − 0.773·49-s + 7.81·53-s − 12.2·55-s + 3.16·59-s + 2·61-s + 13.2·65-s + 3.94·67-s + 10.6·71-s + 10.0·73-s − 10.9·77-s − 14.8·79-s + 6.52·83-s + ⋯ |
| L(s) = 1 | − 1.24·5-s − 0.943·7-s + 1.32·11-s − 1.32·13-s − 0.633·17-s + 0.941·19-s − 0.414·23-s + 0.541·25-s − 0.814·29-s − 0.477·31-s + 1.17·35-s + 0.164·37-s + 0.876·41-s − 1.69·43-s + 1.54·47-s − 0.110·49-s + 1.07·53-s − 1.64·55-s + 0.412·59-s + 0.256·61-s + 1.64·65-s + 0.481·67-s + 1.25·71-s + 1.17·73-s − 1.25·77-s − 1.67·79-s + 0.716·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9194898382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9194898382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 7.81T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.974544123967415573426948347849, −7.989317280141326960668290413573, −7.22853498449560138192863877732, −6.81876087055696558575812390211, −5.82371788929414641018500901306, −4.78158404834047202204560301246, −3.90145258189997707863476773870, −3.41008948629845512181075181318, −2.20082193095866099539715802491, −0.57908031945515831172735156982,
0.57908031945515831172735156982, 2.20082193095866099539715802491, 3.41008948629845512181075181318, 3.90145258189997707863476773870, 4.78158404834047202204560301246, 5.82371788929414641018500901306, 6.81876087055696558575812390211, 7.22853498449560138192863877732, 7.989317280141326960668290413573, 8.974544123967415573426948347849
