| L(s) = 1 | + 4.58·2-s − 1.47·3-s + 12.9·4-s + 19.0·5-s − 6.77·6-s + 28.2·7-s + 22.8·8-s − 24.8·9-s + 87.3·10-s + 55.7·11-s − 19.2·12-s − 15.4·13-s + 129.·14-s − 28.2·15-s + 0.837·16-s + 74.5·17-s − 113.·18-s − 137.·19-s + 247.·20-s − 41.7·21-s + 255.·22-s − 23·23-s − 33.8·24-s + 238.·25-s − 70.9·26-s + 76.6·27-s + 366.·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.284·3-s + 1.62·4-s + 1.70·5-s − 0.461·6-s + 1.52·7-s + 1.01·8-s − 0.918·9-s + 2.76·10-s + 1.52·11-s − 0.462·12-s − 0.330·13-s + 2.46·14-s − 0.485·15-s + 0.0130·16-s + 1.06·17-s − 1.48·18-s − 1.66·19-s + 2.77·20-s − 0.433·21-s + 2.47·22-s − 0.208·23-s − 0.287·24-s + 1.91·25-s − 0.534·26-s + 0.546·27-s + 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.294643299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.294643299\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.58T + 8T^{2} \) |
| 3 | \( 1 + 1.47T + 27T^{2} \) |
| 5 | \( 1 - 19.0T + 125T^{2} \) |
| 7 | \( 1 - 28.2T + 343T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 137.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 238.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 598.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 260.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 637.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 107.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 812.T + 9.12e5T^{2} \) |
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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
−10.47444388531409794706843908390, −9.245805278459181565972353493224, −8.445812036461779514744243790242, −6.91307539183090740280240148419, −6.12206322237009933497465633476, −5.48379082150062185931846096646, −4.85646329662303546620754453496, −3.71496208162689738326032043225, −2.28232809493039435317504343511, −1.58373950586881426277758996604,
1.58373950586881426277758996604, 2.28232809493039435317504343511, 3.71496208162689738326032043225, 4.85646329662303546620754453496, 5.48379082150062185931846096646, 6.12206322237009933497465633476, 6.91307539183090740280240148419, 8.445812036461779514744243790242, 9.245805278459181565972353493224, 10.47444388531409794706843908390
