| L(s) = 1 | + 2.39·3-s + 3.25·5-s + 2.71·9-s − 0.402·11-s − 1.63·13-s + 7.77·15-s + 4.43·17-s + 19-s + 0.837·23-s + 5.57·25-s − 0.686·27-s − 5.33·29-s + 3.16·31-s − 0.962·33-s + 11.4·37-s − 3.89·39-s − 2·41-s + 1.93·43-s + 8.82·45-s + 12.6·47-s + 10.6·51-s − 11.5·53-s − 1.31·55-s + 2.39·57-s + 5.16·59-s + 0.126·61-s − 5.30·65-s + ⋯ |
| L(s) = 1 | + 1.37·3-s + 1.45·5-s + 0.904·9-s − 0.121·11-s − 0.452·13-s + 2.00·15-s + 1.07·17-s + 0.229·19-s + 0.174·23-s + 1.11·25-s − 0.132·27-s − 0.990·29-s + 0.567·31-s − 0.167·33-s + 1.87·37-s − 0.623·39-s − 0.312·41-s + 0.295·43-s + 1.31·45-s + 1.84·47-s + 1.48·51-s − 1.58·53-s − 0.176·55-s + 0.316·57-s + 0.673·59-s + 0.0161·61-s − 0.657·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.869301916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.869301916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 + 0.402T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 23 | \( 1 - 0.837T + 23T^{2} \) |
| 29 | \( 1 + 5.33T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 1.93T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 - 0.126T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 2.17T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 7.50T + 97T^{2} \) |
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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
−7.78519710194102977745620684544, −7.52388425241734352968599974593, −6.46593618180412492942795526900, −5.79059626227410906473786155328, −5.18176454034324390915599974195, −4.19770514933298373942048361717, −3.29207584995394540115068442489, −2.60359025475657129036670078771, −2.05781482670116854294927745620, −1.09802029650983048767998412188,
1.09802029650983048767998412188, 2.05781482670116854294927745620, 2.60359025475657129036670078771, 3.29207584995394540115068442489, 4.19770514933298373942048361717, 5.18176454034324390915599974195, 5.79059626227410906473786155328, 6.46593618180412492942795526900, 7.52388425241734352968599974593, 7.78519710194102977745620684544
