| L(s) = 1 | − 1.63·2-s − 3.34·3-s + 0.681·4-s + 5.47·6-s − 0.857·7-s + 2.15·8-s + 8.17·9-s + 4.79·11-s − 2.27·12-s + 6.58·13-s + 1.40·14-s − 4.89·16-s − 3.97·17-s − 13.3·18-s − 4.58·19-s + 2.86·21-s − 7.85·22-s + 3.30·23-s − 7.21·24-s − 10.7·26-s − 17.2·27-s − 0.584·28-s + 4.20·29-s − 2.87·31-s + 3.70·32-s − 16.0·33-s + 6.50·34-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 1.92·3-s + 0.340·4-s + 2.23·6-s − 0.324·7-s + 0.763·8-s + 2.72·9-s + 1.44·11-s − 0.657·12-s + 1.82·13-s + 0.375·14-s − 1.22·16-s − 0.963·17-s − 3.15·18-s − 1.05·19-s + 0.625·21-s − 1.67·22-s + 0.689·23-s − 1.47·24-s − 2.11·26-s − 3.32·27-s − 0.110·28-s + 0.780·29-s − 0.515·31-s + 0.654·32-s − 2.78·33-s + 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4550606788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4550606788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 7 | \( 1 + 0.857T + 7T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 1.68T + 67T^{2} \) |
| 71 | \( 1 + 0.414T + 71T^{2} \) |
| 73 | \( 1 - 3.74T + 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−9.765162024375902870633067650541, −9.132640953833869090108367136941, −8.313640286166733182722440081692, −7.03177407030310936932096797123, −6.47071200750607615235033042349, −5.94423487459310697533537672811, −4.54532030123688556513598997585, −4.02447753894887252823198519970, −1.56747105545516416601144016673, −0.73282836592017873492878982587,
0.73282836592017873492878982587, 1.56747105545516416601144016673, 4.02447753894887252823198519970, 4.54532030123688556513598997585, 5.94423487459310697533537672811, 6.47071200750607615235033042349, 7.03177407030310936932096797123, 8.313640286166733182722440081692, 9.132640953833869090108367136941, 9.765162024375902870633067650541
