| L(s) = 1 | + 2.90·3-s − 5-s − 7-s + 5.42·9-s + 11-s + 0.903·13-s − 2.90·15-s − 0.903·17-s + 7.05·19-s − 2.90·21-s − 1.37·23-s + 25-s + 7.05·27-s + 3.80·29-s + 0.280·31-s + 2.90·33-s + 35-s + 2.42·37-s + 2.62·39-s + 2.28·41-s − 6.23·43-s − 5.42·45-s + 1.65·47-s + 49-s − 2.62·51-s + 5.18·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s − 0.447·5-s − 0.377·7-s + 1.80·9-s + 0.301·11-s + 0.250·13-s − 0.749·15-s − 0.219·17-s + 1.61·19-s − 0.633·21-s − 0.287·23-s + 0.200·25-s + 1.35·27-s + 0.706·29-s + 0.0504·31-s + 0.505·33-s + 0.169·35-s + 0.399·37-s + 0.419·39-s + 0.356·41-s − 0.950·43-s − 0.809·45-s + 0.241·47-s + 0.142·49-s − 0.367·51-s + 0.712·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.719947367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.719947367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 13 | \( 1 - 0.903T + 13T^{2} \) |
| 17 | \( 1 + 0.903T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 3.80T + 29T^{2} \) |
| 31 | \( 1 - 0.280T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 2.28T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 5.18T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 3.47T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 0.949T + 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 - 6.56T + 97T^{2} \) |
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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
−8.057209001380655624626084648066, −7.54345255353168771875182547560, −6.91717740384698229578317907431, −6.07026281150612239760575800116, −4.98123624785157666626051346755, −4.15258923636992207811699523655, −3.43364469754422910076683166990, −2.97436689197570229911758495835, −2.03695740953935991024980364643, −0.964472788754852069578730334943,
0.964472788754852069578730334943, 2.03695740953935991024980364643, 2.97436689197570229911758495835, 3.43364469754422910076683166990, 4.15258923636992207811699523655, 4.98123624785157666626051346755, 6.07026281150612239760575800116, 6.91717740384698229578317907431, 7.54345255353168771875182547560, 8.057209001380655624626084648066
