| L(s) = 1 | − 1.15·2-s − 3.02·3-s − 0.660·4-s + 3.50·6-s + 1.50·7-s + 3.07·8-s + 6.16·9-s − 5.81·11-s + 2.00·12-s − 5.50·13-s − 1.73·14-s − 2.24·16-s + 0.790·17-s − 7.13·18-s − 3.89·19-s − 4.55·21-s + 6.73·22-s − 5.24·23-s − 9.32·24-s + 6.36·26-s − 9.57·27-s − 0.993·28-s + 6.55·29-s + 31-s − 3.56·32-s + 17.6·33-s − 0.915·34-s + ⋯ |
| L(s) = 1 | − 0.818·2-s − 1.74·3-s − 0.330·4-s + 1.43·6-s + 0.568·7-s + 1.08·8-s + 2.05·9-s − 1.75·11-s + 0.577·12-s − 1.52·13-s − 0.464·14-s − 0.560·16-s + 0.191·17-s − 1.68·18-s − 0.894·19-s − 0.993·21-s + 1.43·22-s − 1.09·23-s − 1.90·24-s + 1.24·26-s − 1.84·27-s − 0.187·28-s + 1.21·29-s + 0.179·31-s − 0.629·32-s + 3.06·33-s − 0.156·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2447406997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2447406997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 - 0.790T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 37 | \( 1 + 0.476T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 2.79T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.13T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 1.81T + 67T^{2} \) |
| 71 | \( 1 - 5.16T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 0.510T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 0.260T + 97T^{2} \) |
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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
−10.16509212952995457649157591354, −10.02602772855090869978158299928, −8.491590734748154854810840081260, −7.72767551908656476348830286801, −6.98611549645941963030043093703, −5.71461897221504461218317548952, −4.97545433602802358636349929294, −4.44835940749758252217913500955, −2.16573780775582346285068568045, −0.48639576675174269073375387096,
0.48639576675174269073375387096, 2.16573780775582346285068568045, 4.44835940749758252217913500955, 4.97545433602802358636349929294, 5.71461897221504461218317548952, 6.98611549645941963030043093703, 7.72767551908656476348830286801, 8.491590734748154854810840081260, 10.02602772855090869978158299928, 10.16509212952995457649157591354
