| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 3.46·7-s + 8-s + 9-s + 10-s + 4.15·11-s + 12-s − 3.73·13-s − 3.46·14-s + 15-s + 16-s + 4.63·17-s + 18-s + 5.88·19-s + 20-s − 3.46·21-s + 4.15·22-s + 4.36·23-s + 24-s + 25-s − 3.73·26-s + 27-s − 3.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.30·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.25·11-s + 0.288·12-s − 1.03·13-s − 0.926·14-s + 0.258·15-s + 0.250·16-s + 1.12·17-s + 0.235·18-s + 1.34·19-s + 0.223·20-s − 0.756·21-s + 0.885·22-s + 0.909·23-s + 0.204·24-s + 0.200·25-s − 0.733·26-s + 0.192·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.174484697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.174484697\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 0.895T + 31T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 7.06T + 97T^{2} \) |
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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
−9.699290933754619218068970629825, −9.353068787179119883412531751851, −8.141137505447247883184167040585, −7.00474803968818219948438586368, −6.62888186479700238174126723391, −5.54672999717015247878686429738, −4.60577403301616761998367369699, −3.29197353545870851404045489794, −2.98693127049187230059033168190, −1.37152531721736187502596462069,
1.37152531721736187502596462069, 2.98693127049187230059033168190, 3.29197353545870851404045489794, 4.60577403301616761998367369699, 5.54672999717015247878686429738, 6.62888186479700238174126723391, 7.00474803968818219948438586368, 8.141137505447247883184167040585, 9.353068787179119883412531751851, 9.699290933754619218068970629825
