| L(s) = 1 | + 2-s + 4-s + 3.52·5-s + 0.783·7-s + 8-s + 3.52·10-s + 3.05·11-s + 1.18·13-s + 0.783·14-s + 16-s − 2.89·17-s − 4.10·19-s + 3.52·20-s + 3.05·22-s + 1.44·23-s + 7.44·25-s + 1.18·26-s + 0.783·28-s − 7.36·29-s + 5.50·31-s + 32-s − 2.89·34-s + 2.76·35-s + 8.23·37-s − 4.10·38-s + 3.52·40-s + 6.93·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.57·5-s + 0.296·7-s + 0.353·8-s + 1.11·10-s + 0.919·11-s + 0.329·13-s + 0.209·14-s + 0.250·16-s − 0.701·17-s − 0.942·19-s + 0.788·20-s + 0.650·22-s + 0.301·23-s + 1.48·25-s + 0.233·26-s + 0.148·28-s − 1.36·29-s + 0.989·31-s + 0.176·32-s − 0.495·34-s + 0.467·35-s + 1.35·37-s − 0.666·38-s + 0.557·40-s + 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.643288107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.643288107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 - 0.783T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 - 8.23T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 + 9.93T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 - 0.890T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 8.73T + 97T^{2} \) |
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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
−8.578971433854534642148669858039, −7.54513439135011770364836415103, −6.65513452629611528145299413905, −6.09901920675675254372080985331, −5.67301806925843634369599993935, −4.60514849687521609900809052982, −4.07103136509923373299958724453, −2.80446667487571441028544539736, −2.08029848719333942689933325264, −1.24289789062449129137653746533,
1.24289789062449129137653746533, 2.08029848719333942689933325264, 2.80446667487571441028544539736, 4.07103136509923373299958724453, 4.60514849687521609900809052982, 5.67301806925843634369599993935, 6.09901920675675254372080985331, 6.65513452629611528145299413905, 7.54513439135011770364836415103, 8.578971433854534642148669858039
