| L(s) = 1 | + 1.43·2-s − 5.93·4-s − 5·5-s − 20.0·8-s − 7.19·10-s − 7.61·11-s − 52.3·13-s + 18.6·16-s − 49.7·17-s − 140.·19-s + 29.6·20-s − 10.9·22-s + 23.4·23-s + 25·25-s − 75.3·26-s − 157.·29-s − 127.·31-s + 187.·32-s − 71.5·34-s − 115.·37-s − 202.·38-s + 100.·40-s − 188.·41-s + 322.·43-s + 45.1·44-s + 33.7·46-s − 76.6·47-s + ⋯ |
| L(s) = 1 | + 0.508·2-s − 0.741·4-s − 0.447·5-s − 0.885·8-s − 0.227·10-s − 0.208·11-s − 1.11·13-s + 0.290·16-s − 0.709·17-s − 1.69·19-s + 0.331·20-s − 0.106·22-s + 0.212·23-s + 0.200·25-s − 0.568·26-s − 1.00·29-s − 0.740·31-s + 1.03·32-s − 0.360·34-s − 0.513·37-s − 0.863·38-s + 0.396·40-s − 0.718·41-s + 1.14·43-s + 0.154·44-s + 0.108·46-s − 0.237·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4680016709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4680016709\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.43T + 8T^{2} \) |
| 11 | \( 1 + 7.61T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 76.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 107.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 915.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 451.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 907.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 755.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 22.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 549.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.887385867191743533014129876072, −7.914205089387325300040572876098, −7.17340154100322498602260083793, −6.23596249848109304354242060448, −5.36957035331700971908378455498, −4.56210057169494086381435836876, −4.04011758013969072746700681079, −3.00212000890085459665972406671, −1.97605024735257376920726237790, −0.27153326490052785236700977258,
0.27153326490052785236700977258, 1.97605024735257376920726237790, 3.00212000890085459665972406671, 4.04011758013969072746700681079, 4.56210057169494086381435836876, 5.36957035331700971908378455498, 6.23596249848109304354242060448, 7.17340154100322498602260083793, 7.914205089387325300040572876098, 8.887385867191743533014129876072
