| L(s) = 1 | − 3-s + 3.80·5-s + 5.13·7-s + 9-s + 0.334·11-s − 3.80·15-s + 4.13·17-s − 5.94·19-s − 5.13·21-s + 0.334·23-s + 9.47·25-s − 27-s − 0.195·29-s + 4.80·31-s − 0.334·33-s + 19.5·35-s − 2.13·37-s + 3.46·41-s − 2.86·43-s + 3.80·45-s − 3.66·47-s + 19.4·49-s − 4.13·51-s + 9.41·53-s + 1.27·55-s + 5.94·57-s − 6.27·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.70·5-s + 1.94·7-s + 0.333·9-s + 0.100·11-s − 0.982·15-s + 1.00·17-s − 1.36·19-s − 1.12·21-s + 0.0698·23-s + 1.89·25-s − 0.192·27-s − 0.0363·29-s + 0.862·31-s − 0.0582·33-s + 3.30·35-s − 0.351·37-s + 0.541·41-s − 0.436·43-s + 0.567·45-s − 0.534·47-s + 2.77·49-s − 0.579·51-s + 1.29·53-s + 0.171·55-s + 0.787·57-s − 0.817·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.027991422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.027991422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 0.334T + 11T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 - 0.334T + 23T^{2} \) |
| 29 | \( 1 + 0.195T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 + 2.13T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 + 6.27T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.33T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 0.134T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 0.390T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.457205757890769230617942740185, −7.77034771402285874982462925743, −6.81576699904175026043821434328, −6.07252424108368229405500390543, −5.43439086495657568602450961003, −4.92123508138568010411649881347, −4.14042782948675432263032050991, −2.59698633492082307489630334788, −1.79016835378878497581560171697, −1.16841179027851472316925511544,
1.16841179027851472316925511544, 1.79016835378878497581560171697, 2.59698633492082307489630334788, 4.14042782948675432263032050991, 4.92123508138568010411649881347, 5.43439086495657568602450961003, 6.07252424108368229405500390543, 6.81576699904175026043821434328, 7.77034771402285874982462925743, 8.457205757890769230617942740185
