| L(s) = 1 | − 4.11·2-s + 9.51·3-s + 8.93·4-s + 14.5·5-s − 39.1·6-s + 21.0·7-s − 3.83·8-s + 63.5·9-s − 59.7·10-s − 11·11-s + 84.9·12-s − 13·13-s − 86.5·14-s + 138.·15-s − 55.6·16-s − 67.8·17-s − 261.·18-s − 88.0·19-s + 129.·20-s + 200.·21-s + 45.2·22-s − 118.·23-s − 36.5·24-s + 85.8·25-s + 53.4·26-s + 347.·27-s + 187.·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.83·3-s + 1.11·4-s + 1.29·5-s − 2.66·6-s + 1.13·7-s − 0.169·8-s + 2.35·9-s − 1.88·10-s − 0.301·11-s + 2.04·12-s − 0.277·13-s − 1.65·14-s + 2.37·15-s − 0.869·16-s − 0.967·17-s − 3.42·18-s − 1.06·19-s + 1.45·20-s + 2.08·21-s + 0.438·22-s − 1.07·23-s − 0.310·24-s + 0.686·25-s + 0.403·26-s + 2.47·27-s + 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.936560427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936560427\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 4.11T + 8T^{2} \) |
| 3 | \( 1 - 9.51T + 27T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 7 | \( 1 - 21.0T + 343T^{2} \) |
| 17 | \( 1 + 67.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 372.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 230.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 486.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 556.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 479.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 491.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 563.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 431.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 171.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−13.03885090943274335656718934895, −11.09997429899024740832627510480, −9.959005646988513214168075875406, −9.512454550634819756266142578460, −8.349712755932882296983791598321, −8.085614172721866158614436374236, −6.68068669037449947650413042824, −4.53581222366034707719847456718, −2.31965205367188452341099535766, −1.75221968150182339193807405856,
1.75221968150182339193807405856, 2.31965205367188452341099535766, 4.53581222366034707719847456718, 6.68068669037449947650413042824, 8.085614172721866158614436374236, 8.349712755932882296983791598321, 9.512454550634819756266142578460, 9.959005646988513214168075875406, 11.09997429899024740832627510480, 13.03885090943274335656718934895
