L(s) = 1 | + 3.60·2-s − 0.812·3-s + 5.00·4-s + 5.99·5-s − 2.92·6-s − 26.3·7-s − 10.8·8-s − 26.3·9-s + 21.6·10-s − 27.0·11-s − 4.06·12-s + 83.2·13-s − 95.0·14-s − 4.87·15-s − 78.9·16-s + 19.3·17-s − 94.9·18-s − 1.45·19-s + 30.0·20-s + 21.4·21-s − 97.7·22-s + 56.8·23-s + 8.77·24-s − 89.0·25-s + 300.·26-s + 43.3·27-s − 131.·28-s + ⋯ |
L(s) = 1 | + 1.27·2-s − 0.156·3-s + 0.625·4-s + 0.536·5-s − 0.199·6-s − 1.42·7-s − 0.477·8-s − 0.975·9-s + 0.683·10-s − 0.742·11-s − 0.0977·12-s + 1.77·13-s − 1.81·14-s − 0.0838·15-s − 1.23·16-s + 0.276·17-s − 1.24·18-s − 0.0175·19-s + 0.335·20-s + 0.222·21-s − 0.946·22-s + 0.515·23-s + 0.0746·24-s − 0.712·25-s + 2.26·26-s + 0.308·27-s − 0.890·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.724377888\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.724377888\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 3.60T + 8T^{2} \) |
| 3 | \( 1 + 0.812T + 27T^{2} \) |
| 5 | \( 1 - 5.99T + 125T^{2} \) |
| 7 | \( 1 + 26.3T + 343T^{2} \) |
| 11 | \( 1 + 27.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.45T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 65.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 164.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 111.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 637.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 810.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 789.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 731.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 432.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 45.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 942.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 355.T + 9.12e5T^{2} \) |
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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
−8.841211972598187632029467831859, −8.244239162480887307748573235390, −6.70495361055913348450150023000, −6.29292771178275166679174796623, −5.66037749317872023192247166049, −4.99886675206753548697461176362, −3.65309345609232363198307728890, −3.26896617020922453600995618822, −2.34684517554516375725618121499, −0.61227447551952609622015296602,
0.61227447551952609622015296602, 2.34684517554516375725618121499, 3.26896617020922453600995618822, 3.65309345609232363198307728890, 4.99886675206753548697461176362, 5.66037749317872023192247166049, 6.29292771178275166679174796623, 6.70495361055913348450150023000, 8.244239162480887307748573235390, 8.841211972598187632029467831859