| L(s) = 1 | − 3-s + 3.37·5-s − 2.52·7-s + 9-s − 5.84·13-s − 3.37·15-s + 2.67·17-s + 0.939·19-s + 2.52·21-s − 2·23-s + 6.37·25-s − 27-s − 0.792·29-s − 1.62·31-s − 8.51·35-s + 5·37-s + 5.84·39-s − 10.8·41-s + 6.63·43-s + 3.37·45-s + 12.7·47-s − 0.627·49-s − 2.67·51-s − 4.11·53-s − 0.939·57-s + 6·59-s + 5.98·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.50·5-s − 0.954·7-s + 0.333·9-s − 1.61·13-s − 0.870·15-s + 0.648·17-s + 0.215·19-s + 0.550·21-s − 0.417·23-s + 1.27·25-s − 0.192·27-s − 0.147·29-s − 0.292·31-s − 1.43·35-s + 0.821·37-s + 0.935·39-s − 1.70·41-s + 1.01·43-s + 0.502·45-s + 1.85·47-s − 0.0896·49-s − 0.374·51-s − 0.565·53-s − 0.124·57-s + 0.781·59-s + 0.766·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635841858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635841858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 0.939T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 0.792T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 9.15T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 0.627T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.994840830851206310605120703302, −7.16822063539921671921528534320, −6.62740816480593185388778281681, −5.87103594943524118727667302022, −5.42650833260089242818823930803, −4.71760084824913737032485732576, −3.59538272052619371057867521364, −2.60874016544420432367173060961, −1.96870042560191614500153088363, −0.67920285723717300731859359705,
0.67920285723717300731859359705, 1.96870042560191614500153088363, 2.60874016544420432367173060961, 3.59538272052619371057867521364, 4.71760084824913737032485732576, 5.42650833260089242818823930803, 5.87103594943524118727667302022, 6.62740816480593185388778281681, 7.16822063539921671921528534320, 7.994840830851206310605120703302
