| L(s) = 1 | + 2·2-s − 7.50·3-s + 4·4-s − 15.0·6-s + 17.5·7-s + 8·8-s + 29.3·9-s + 48.7·11-s − 30.0·12-s + 81.1·13-s + 35.1·14-s + 16·16-s − 35.2·17-s + 58.6·18-s + 43.9·19-s − 131.·21-s + 97.4·22-s + 92.2·23-s − 60.0·24-s + 162.·26-s − 17.2·27-s + 70.2·28-s − 287.·29-s + 117.·31-s + 32·32-s − 365.·33-s − 70.5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.44·3-s + 0.5·4-s − 1.02·6-s + 0.947·7-s + 0.353·8-s + 1.08·9-s + 1.33·11-s − 0.722·12-s + 1.73·13-s + 0.670·14-s + 0.250·16-s − 0.503·17-s + 0.767·18-s + 0.530·19-s − 1.36·21-s + 0.944·22-s + 0.836·23-s − 0.510·24-s + 1.22·26-s − 0.123·27-s + 0.473·28-s − 1.84·29-s + 0.679·31-s + 0.176·32-s − 1.92·33-s − 0.355·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.901253569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901253569\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 7.50T + 27T^{2} \) |
| 7 | \( 1 - 17.5T + 343T^{2} \) |
| 11 | \( 1 - 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 81.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 68.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 19.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 708.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 144.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 195.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 524.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 457.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 840.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−9.344261537279629119427202153754, −8.508134670616547598330311823908, −7.35602412091103875926760886207, −6.49740431784361667834951688106, −5.95008164330450704916013889676, −5.15250086829130313003407651366, −4.32979163338707636387513928710, −3.50430309154144461483543205643, −1.68754732245608658036118234769, −0.929658944815369527419241202194,
0.929658944815369527419241202194, 1.68754732245608658036118234769, 3.50430309154144461483543205643, 4.32979163338707636387513928710, 5.15250086829130313003407651366, 5.95008164330450704916013889676, 6.49740431784361667834951688106, 7.35602412091103875926760886207, 8.508134670616547598330311823908, 9.344261537279629119427202153754
