| L(s) = 1 | + 2-s + 2.32·3-s + 4-s − 0.295·5-s + 2.32·6-s − 5.17·7-s + 8-s + 2.41·9-s − 0.295·10-s + 1.37·11-s + 2.32·12-s + 5.08·13-s − 5.17·14-s − 0.689·15-s + 16-s + 5.70·17-s + 2.41·18-s + 2.67·19-s − 0.295·20-s − 12.0·21-s + 1.37·22-s + 7.13·23-s + 2.32·24-s − 4.91·25-s + 5.08·26-s − 1.35·27-s − 5.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 0.5·4-s − 0.132·5-s + 0.950·6-s − 1.95·7-s + 0.353·8-s + 0.806·9-s − 0.0935·10-s + 0.415·11-s + 0.672·12-s + 1.40·13-s − 1.38·14-s − 0.177·15-s + 0.250·16-s + 1.38·17-s + 0.570·18-s + 0.613·19-s − 0.0661·20-s − 2.63·21-s + 0.293·22-s + 1.48·23-s + 0.475·24-s − 0.982·25-s + 0.996·26-s − 0.260·27-s − 0.978·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.703764625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.703764625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 0.295T + 5T^{2} \) |
| 7 | \( 1 + 5.17T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
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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
−8.132737740742385724621536442649, −7.14981010276059133553870828457, −6.83338093596587509686337011907, −5.88327435511078164801392236156, −5.36125240474628470904357274980, −3.85181332686884620870722500937, −3.47119064658165632864279534784, −3.26460377350059946775940186626, −2.21122606867350746853137294895, −0.985252499985261254933561891822,
0.985252499985261254933561891822, 2.21122606867350746853137294895, 3.26460377350059946775940186626, 3.47119064658165632864279534784, 3.85181332686884620870722500937, 5.36125240474628470904357274980, 5.88327435511078164801392236156, 6.83338093596587509686337011907, 7.14981010276059133553870828457, 8.132737740742385724621536442649
