| L(s) = 1 | + 0.792·2-s − 1.37·4-s + 2.52·5-s + 7-s − 2.67·8-s + 2·10-s + 3.46·11-s − 13-s + 0.792·14-s + 0.627·16-s + 1.58·17-s + 1.62·19-s − 3.46·20-s + 2.74·22-s + 4.40·23-s + 1.37·25-s − 0.792·26-s − 1.37·28-s − 5.98·29-s + 6.37·31-s + 5.84·32-s + 1.25·34-s + 2.52·35-s + 6.74·37-s + 1.28·38-s − 6.74·40-s − 1.58·41-s + ⋯ |
| L(s) = 1 | + 0.560·2-s − 0.686·4-s + 1.12·5-s + 0.377·7-s − 0.944·8-s + 0.632·10-s + 1.04·11-s − 0.277·13-s + 0.211·14-s + 0.156·16-s + 0.384·17-s + 0.373·19-s − 0.774·20-s + 0.585·22-s + 0.918·23-s + 0.274·25-s − 0.155·26-s − 0.259·28-s − 1.11·29-s + 1.14·31-s + 1.03·32-s + 0.215·34-s + 0.426·35-s + 1.10·37-s + 0.209·38-s − 1.06·40-s − 0.247·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221427981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221427981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 6.37T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 + 0.939T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8.74T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 2.52T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−9.954707534196256647131430096300, −9.454866055208150250603768943460, −8.786522129115983082414957876230, −7.66446771099370676170080181524, −6.45345746824596232353535315252, −5.72976019489439689032558988227, −4.91732876589398239976105394569, −3.97169503762111151335883298448, −2.75758820417920982148698911055, −1.28002645204984240287764115096,
1.28002645204984240287764115096, 2.75758820417920982148698911055, 3.97169503762111151335883298448, 4.91732876589398239976105394569, 5.72976019489439689032558988227, 6.45345746824596232353535315252, 7.66446771099370676170080181524, 8.786522129115983082414957876230, 9.454866055208150250603768943460, 9.954707534196256647131430096300
