| L(s) = 1 | − 2-s − 0.732·3-s + 4-s + 5-s + 0.732·6-s + 4.46·7-s − 8-s − 2.46·9-s − 10-s + 2.73·11-s − 0.732·12-s − 2.46·13-s − 4.46·14-s − 0.732·15-s + 16-s + 3.26·17-s + 2.46·18-s − 6.46·19-s + 20-s − 3.26·21-s − 2.73·22-s + 8.19·23-s + 0.732·24-s + 25-s + 2.46·26-s + 4·27-s + 4.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s + 0.447·5-s + 0.298·6-s + 1.68·7-s − 0.353·8-s − 0.821·9-s − 0.316·10-s + 0.823·11-s − 0.211·12-s − 0.683·13-s − 1.19·14-s − 0.189·15-s + 0.250·16-s + 0.792·17-s + 0.580·18-s − 1.48·19-s + 0.223·20-s − 0.713·21-s − 0.582·22-s + 1.70·23-s + 0.149·24-s + 0.200·25-s + 0.483·26-s + 0.769·27-s + 0.843·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.096260444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096260444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 9.73T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−10.99961431841786874305993870716, −10.50535292440714035812457313933, −9.152498231602948523777427627264, −8.566197262430930748098909763116, −7.60968912236829993084843837988, −6.52893355895303371305281446852, −5.47166556350359167234355663510, −4.54462659814139234528543846504, −2.61463251679631140882121102810, −1.25235821422080383469071644201,
1.25235821422080383469071644201, 2.61463251679631140882121102810, 4.54462659814139234528543846504, 5.47166556350359167234355663510, 6.52893355895303371305281446852, 7.60968912236829993084843837988, 8.566197262430930748098909763116, 9.152498231602948523777427627264, 10.50535292440714035812457313933, 10.99961431841786874305993870716
