| L(s) = 1 | + 0.703·2-s − 1.64·3-s − 1.50·4-s − 1.15·6-s + 0.501·7-s − 2.46·8-s − 0.308·9-s − 1.80·11-s + 2.47·12-s + 1.63·13-s + 0.352·14-s + 1.27·16-s + 1.92·17-s − 0.216·18-s − 0.869·19-s − 0.822·21-s − 1.26·22-s + 4.86·23-s + 4.04·24-s + 1.14·26-s + 5.42·27-s − 0.754·28-s − 3.14·29-s + 2.39·31-s + 5.82·32-s + 2.95·33-s + 1.35·34-s + ⋯ |
| L(s) = 1 | + 0.497·2-s − 0.947·3-s − 0.752·4-s − 0.470·6-s + 0.189·7-s − 0.871·8-s − 0.102·9-s − 0.543·11-s + 0.713·12-s + 0.452·13-s + 0.0942·14-s + 0.319·16-s + 0.467·17-s − 0.0510·18-s − 0.199·19-s − 0.179·21-s − 0.270·22-s + 1.01·23-s + 0.825·24-s + 0.225·26-s + 1.04·27-s − 0.142·28-s − 0.584·29-s + 0.429·31-s + 1.03·32-s + 0.515·33-s + 0.232·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004526529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004526529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.703T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 7 | \( 1 - 0.501T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 0.869T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 6.06T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 - 7.44T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 - 5.16T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−10.22979828345910195192678776573, −9.213292283685418159901567970935, −8.485774625190610602080423167931, −7.49051030146023695820223890617, −6.27875172398799882504508878872, −5.58477583567624830901314199711, −4.95499689055015606710689620319, −3.97660508650716889292384729834, −2.79713889881529168224706571123, −0.77691052721546439621490319941,
0.77691052721546439621490319941, 2.79713889881529168224706571123, 3.97660508650716889292384729834, 4.95499689055015606710689620319, 5.58477583567624830901314199711, 6.27875172398799882504508878872, 7.49051030146023695820223890617, 8.485774625190610602080423167931, 9.213292283685418159901567970935, 10.22979828345910195192678776573
