| L(s) = 1 | − 1.60·2-s − 3-s + 0.580·4-s + 1.60·6-s + 2.35·7-s + 2.28·8-s + 9-s + 1.21·11-s − 0.580·12-s − 1.28·13-s − 3.78·14-s − 4.82·16-s + 2.46·17-s − 1.60·18-s − 0.168·19-s − 2.35·21-s − 1.95·22-s + 8.96·23-s − 2.28·24-s + 2.06·26-s − 27-s + 1.36·28-s − 2.39·29-s + 6.32·31-s + 3.18·32-s − 1.21·33-s − 3.95·34-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.290·4-s + 0.655·6-s + 0.889·7-s + 0.806·8-s + 0.333·9-s + 0.367·11-s − 0.167·12-s − 0.357·13-s − 1.01·14-s − 1.20·16-s + 0.596·17-s − 0.378·18-s − 0.0387·19-s − 0.513·21-s − 0.417·22-s + 1.86·23-s − 0.465·24-s + 0.405·26-s − 0.192·27-s + 0.258·28-s − 0.445·29-s + 1.13·31-s + 0.563·32-s − 0.212·33-s − 0.677·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9555305332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9555305332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 + 0.168T + 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 9.44T + 37T^{2} \) |
| 41 | \( 1 + 1.59T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + 6.83T + 71T^{2} \) |
| 73 | \( 1 - 0.476T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.585331634646296506255689917975, −7.911278649546010151115464070912, −7.28870662992269990079542534900, −6.59238934157754131911511728379, −5.49798366282892250866583258050, −4.81670636926919321719665721766, −4.13875525829747253351557047656, −2.74115732177549976531121145979, −1.50386353288179550815880034017, −0.78738399647374960853071309375,
0.78738399647374960853071309375, 1.50386353288179550815880034017, 2.74115732177549976531121145979, 4.13875525829747253351557047656, 4.81670636926919321719665721766, 5.49798366282892250866583258050, 6.59238934157754131911511728379, 7.28870662992269990079542534900, 7.911278649546010151115464070912, 8.585331634646296506255689917975
