| L(s) = 1 | − 2.49·2-s + 3-s + 4.23·4-s − 2.49·6-s − 5.58·8-s + 9-s − 4.47·11-s + 4.23·12-s + 5.23·13-s + 5.47·16-s + 0.763·17-s − 2.49·18-s − 8.08·19-s + 11.1·22-s − 3.08·23-s − 5.58·24-s − 13.0·26-s + 27-s + 2·29-s + 1.90·31-s − 2.49·32-s − 4.47·33-s − 1.90·34-s + 4.23·36-s + 6.17·37-s + 20.1·38-s + 5.23·39-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.11·4-s − 1.01·6-s − 1.97·8-s + 0.333·9-s − 1.34·11-s + 1.22·12-s + 1.45·13-s + 1.36·16-s + 0.185·17-s − 0.588·18-s − 1.85·19-s + 2.38·22-s − 0.643·23-s − 1.13·24-s − 2.56·26-s + 0.192·27-s + 0.371·29-s + 0.342·31-s − 0.441·32-s − 0.778·33-s − 0.327·34-s + 0.706·36-s + 1.01·37-s + 3.27·38-s + 0.838·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8348962869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8348962869\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 8.08T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 6.17T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 - 0.472T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.516740688853783236487606568411, −8.049257355378030041786881098015, −7.50813997938853656851344633176, −6.45653265233016072582536806502, −6.03635222696970332372853711043, −4.67282856612736425552398964030, −3.57778720919361699044159201686, −2.53121220766337814828883317375, −1.88981751599083972301318138514, −0.66319198782652543156763378256,
0.66319198782652543156763378256, 1.88981751599083972301318138514, 2.53121220766337814828883317375, 3.57778720919361699044159201686, 4.67282856612736425552398964030, 6.03635222696970332372853711043, 6.45653265233016072582536806502, 7.50813997938853656851344633176, 8.049257355378030041786881098015, 8.516740688853783236487606568411
