| L(s) = 1 | − 2.37·2-s + 0.742·3-s + 3.62·4-s − 1.76·6-s − 2.43·7-s − 3.84·8-s − 2.44·9-s − 5.52·11-s + 2.69·12-s − 5.05·13-s + 5.77·14-s + 1.87·16-s + 4.60·17-s + 5.80·18-s + 3.43·19-s − 1.80·21-s + 13.0·22-s − 1.11·23-s − 2.85·24-s + 11.9·26-s − 4.04·27-s − 8.81·28-s − 2.19·29-s + 2.13·31-s + 3.24·32-s − 4.10·33-s − 10.9·34-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.428·3-s + 1.81·4-s − 0.718·6-s − 0.920·7-s − 1.35·8-s − 0.816·9-s − 1.66·11-s + 0.776·12-s − 1.40·13-s + 1.54·14-s + 0.468·16-s + 1.11·17-s + 1.36·18-s + 0.788·19-s − 0.394·21-s + 2.79·22-s − 0.232·23-s − 0.583·24-s + 2.35·26-s − 0.778·27-s − 1.66·28-s − 0.406·29-s + 0.383·31-s + 0.574·32-s − 0.713·33-s − 1.87·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2554293469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2554293469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 0.742T + 3T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 - 0.531T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.342066244279621217025296440401, −8.009185221775903318059230769981, −7.40520806314009469075532673134, −6.68246620521139948810460778408, −5.63417104202746454532976292512, −4.98766094515326284522248786675, −3.23572873990678965205076773251, −2.84146514675554883783588101470, −1.91643500218561926349165103411, −0.34411797561663992282557146185,
0.34411797561663992282557146185, 1.91643500218561926349165103411, 2.84146514675554883783588101470, 3.23572873990678965205076773251, 4.98766094515326284522248786675, 5.63417104202746454532976292512, 6.68246620521139948810460778408, 7.40520806314009469075532673134, 8.009185221775903318059230769981, 8.342066244279621217025296440401
