| L(s) = 1 | + 0.470·2-s + 2.24·3-s − 1.77·4-s + 0.529·5-s + 1.05·6-s − 7-s − 1.77·8-s + 2.05·9-s + 0.249·10-s − 2.24·11-s − 4.00·12-s + 13-s − 0.470·14-s + 1.19·15-s + 2.71·16-s − 1.30·17-s + 0.968·18-s − 1.47·19-s − 0.941·20-s − 2.24·21-s − 1.05·22-s + 5.83·23-s − 4.00·24-s − 4.71·25-s + 0.470·26-s − 2.11·27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 1.29·3-s − 0.889·4-s + 0.236·5-s + 0.432·6-s − 0.377·7-s − 0.628·8-s + 0.686·9-s + 0.0787·10-s − 0.678·11-s − 1.15·12-s + 0.277·13-s − 0.125·14-s + 0.307·15-s + 0.679·16-s − 0.317·17-s + 0.228·18-s − 0.337·19-s − 0.210·20-s − 0.490·21-s − 0.225·22-s + 1.21·23-s − 0.816·24-s − 0.943·25-s + 0.0923·26-s − 0.407·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315528665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315528665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + 7.02T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−13.92878191226870637036343378418, −13.34452425960702938978586659749, −12.48136097530947487315393465072, −10.64055173704041571584458917132, −9.345127097157821584327934150744, −8.764766573874037691399016881034, −7.55210189783927525293762275792, −5.71590372024041195022493422624, −4.12066848872037228013527344962, −2.79185347271348457784021644014,
2.79185347271348457784021644014, 4.12066848872037228013527344962, 5.71590372024041195022493422624, 7.55210189783927525293762275792, 8.764766573874037691399016881034, 9.345127097157821584327934150744, 10.64055173704041571584458917132, 12.48136097530947487315393465072, 13.34452425960702938978586659749, 13.92878191226870637036343378418
