| L(s) = 1 | + 2.09·2-s + 2.89·3-s + 2.38·4-s − 3.61·5-s + 6.05·6-s + 0.264·7-s + 0.802·8-s + 5.36·9-s − 7.55·10-s + 11-s + 6.89·12-s + 5.66·13-s + 0.554·14-s − 10.4·15-s − 3.08·16-s + 0.577·17-s + 11.2·18-s − 8.60·20-s + 0.766·21-s + 2.09·22-s + 2.58·23-s + 2.32·24-s + 8.03·25-s + 11.8·26-s + 6.84·27-s + 0.631·28-s + 7.43·29-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.67·3-s + 1.19·4-s − 1.61·5-s + 2.47·6-s + 0.100·7-s + 0.283·8-s + 1.78·9-s − 2.39·10-s + 0.301·11-s + 1.98·12-s + 1.57·13-s + 0.148·14-s − 2.69·15-s − 0.771·16-s + 0.140·17-s + 2.64·18-s − 1.92·20-s + 0.167·21-s + 0.446·22-s + 0.539·23-s + 0.473·24-s + 1.60·25-s + 2.32·26-s + 1.31·27-s + 0.119·28-s + 1.38·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.654371882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.654371882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 - 0.264T + 7T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 17 | \( 1 - 0.577T + 17T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 - 0.639T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + 4.15T + 47T^{2} \) |
| 53 | \( 1 + 5.18T + 53T^{2} \) |
| 59 | \( 1 - 4.63T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 - 2.42T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.243261883665974773754177397806, −7.956910211790867722598164137454, −6.85074093403001164802849937724, −6.42695095606757021911975160421, −5.07617787408961151208254331274, −4.24941601220466938477238866653, −3.87141055446800545503681276421, −3.21101500119157364240657357836, −2.67308558508209002441985888390, −1.20508341376257433977586716432,
1.20508341376257433977586716432, 2.67308558508209002441985888390, 3.21101500119157364240657357836, 3.87141055446800545503681276421, 4.24941601220466938477238866653, 5.07617787408961151208254331274, 6.42695095606757021911975160421, 6.85074093403001164802849937724, 7.956910211790867722598164137454, 8.243261883665974773754177397806
