| L(s) = 1 | + 2-s + 1.65·3-s + 4-s + 0.725·5-s + 1.65·6-s + 1.29·7-s + 8-s − 0.269·9-s + 0.725·10-s − 3.71·11-s + 1.65·12-s + 2.30·13-s + 1.29·14-s + 1.19·15-s + 16-s + 6.32·17-s − 0.269·18-s + 0.0212·19-s + 0.725·20-s + 2.14·21-s − 3.71·22-s − 1.30·23-s + 1.65·24-s − 4.47·25-s + 2.30·26-s − 5.40·27-s + 1.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.954·3-s + 0.5·4-s + 0.324·5-s + 0.674·6-s + 0.491·7-s + 0.353·8-s − 0.0896·9-s + 0.229·10-s − 1.12·11-s + 0.477·12-s + 0.640·13-s + 0.347·14-s + 0.309·15-s + 0.250·16-s + 1.53·17-s − 0.0634·18-s + 0.00487·19-s + 0.162·20-s + 0.468·21-s − 0.792·22-s − 0.272·23-s + 0.337·24-s − 0.894·25-s + 0.452·26-s − 1.03·27-s + 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.039392060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.039392060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 5 | \( 1 - 0.725T + 5T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 0.0212T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 7.25T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 - 4.52T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 + 0.643T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 - 0.418T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−10.85470600500216461815499180265, −9.991636738218995253077395919556, −8.972093888940116060710189778839, −7.955089796853058772826087668280, −7.50622136560456556881854933257, −5.89214041651080648345770327155, −5.33500341220536760231560600725, −3.90074527201998946004366696834, −2.98345055186004453631441149281, −1.86331627140814674766273129677,
1.86331627140814674766273129677, 2.98345055186004453631441149281, 3.90074527201998946004366696834, 5.33500341220536760231560600725, 5.89214041651080648345770327155, 7.50622136560456556881854933257, 7.955089796853058772826087668280, 8.972093888940116060710189778839, 9.991636738218995253077395919556, 10.85470600500216461815499180265
