| L(s) = 1 | − 3·3-s − 8·4-s + 5·5-s + 7·7-s + 9·9-s + 42·11-s + 24·12-s + 20·13-s − 15·15-s + 64·16-s + 66·17-s + 38·19-s − 40·20-s − 21·21-s + 12·23-s + 25·25-s − 27·27-s − 56·28-s − 258·29-s + 146·31-s − 126·33-s + 35·35-s − 72·36-s + 434·37-s − 60·39-s − 282·41-s + 20·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.15·11-s + 0.577·12-s + 0.426·13-s − 0.258·15-s + 16-s + 0.941·17-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.108·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.65·29-s + 0.845·31-s − 0.664·33-s + 0.169·35-s − 1/3·36-s + 1.92·37-s − 0.246·39-s − 1.07·41-s + 0.0709·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.261008627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261008627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 - 146 T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 336 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 682 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 810 T + p^{3} T^{2} \) |
| 73 | \( 1 + 124 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1208 T + p^{3} T^{2} \) |
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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
−13.33999715747491387220601626537, −12.25507757611512237536921989041, −11.24148326992387271744824846264, −9.917697286121603683278694291785, −9.140012825049310871337886399587, −7.81787339196846720357315904867, −6.21817264354172821605570724071, −5.13556761134837335702729587063, −3.81319902620835772775865421004, −1.14789243311335868174781089344,
1.14789243311335868174781089344, 3.81319902620835772775865421004, 5.13556761134837335702729587063, 6.21817264354172821605570724071, 7.81787339196846720357315904867, 9.140012825049310871337886399587, 9.917697286121603683278694291785, 11.24148326992387271744824846264, 12.25507757611512237536921989041, 13.33999715747491387220601626537
