L(s) = 1 | + 0.690·3-s + 3.23·5-s − 2.52·9-s + 11-s + 4.83·13-s + 2.23·15-s + 7.46·17-s − 5.88·19-s − 1.20·23-s + 5.49·25-s − 3.81·27-s + 5.24·29-s + 9.40·31-s + 0.690·33-s − 7.03·37-s + 3.33·39-s − 1.05·41-s − 3.24·43-s − 8.17·45-s + 8.87·47-s + 5.15·51-s − 5.25·53-s + 3.23·55-s − 4.06·57-s + 14.5·59-s + 13.7·61-s + 15.6·65-s + ⋯ |
L(s) = 1 | + 0.398·3-s + 1.44·5-s − 0.841·9-s + 0.301·11-s + 1.34·13-s + 0.577·15-s + 1.80·17-s − 1.35·19-s − 0.250·23-s + 1.09·25-s − 0.733·27-s + 0.974·29-s + 1.68·31-s + 0.120·33-s − 1.15·37-s + 0.534·39-s − 0.164·41-s − 0.495·43-s − 1.21·45-s + 1.29·47-s + 0.721·51-s − 0.721·53-s + 0.436·55-s − 0.538·57-s + 1.89·59-s + 1.75·61-s + 1.94·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.602662852\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.602662852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 0.690T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 - 8.87T + 47T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 - 1.20T + 97T^{2} \) |
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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
−8.029750564610338014667986192125, −6.91034309981727021810405246674, −6.20007071886451721406463375336, −5.86265482574410730007131683045, −5.21290088899990324794379686558, −4.16535830773451280050385447024, −3.31776583618293392384050892879, −2.64357914170096295230409312695, −1.78341618307804800659950208829, −0.959312128265039974239357190234,
0.959312128265039974239357190234, 1.78341618307804800659950208829, 2.64357914170096295230409312695, 3.31776583618293392384050892879, 4.16535830773451280050385447024, 5.21290088899990324794379686558, 5.86265482574410730007131683045, 6.20007071886451721406463375336, 6.91034309981727021810405246674, 8.029750564610338014667986192125