| L(s) = 1 | − 4.60·2-s − 3·3-s + 13.2·4-s + 13.8·6-s − 15.5·7-s − 24.1·8-s + 9·9-s + 7.26·11-s − 39.6·12-s + 13·13-s + 71.4·14-s + 5.22·16-s + 27.0·17-s − 41.4·18-s + 73.2·19-s + 46.5·21-s − 33.4·22-s − 198.·23-s + 72.3·24-s − 59.9·26-s − 27·27-s − 205.·28-s + 210.·29-s − 4.60·31-s + 168.·32-s − 21.7·33-s − 124.·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.65·4-s + 0.940·6-s − 0.837·7-s − 1.06·8-s + 0.333·9-s + 0.199·11-s − 0.954·12-s + 0.277·13-s + 1.36·14-s + 0.0815·16-s + 0.385·17-s − 0.543·18-s + 0.884·19-s + 0.483·21-s − 0.324·22-s − 1.79·23-s + 0.615·24-s − 0.451·26-s − 0.192·27-s − 1.38·28-s + 1.34·29-s − 0.0266·31-s + 0.932·32-s − 0.114·33-s − 0.628·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4906046026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4906046026\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 4.60T + 8T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 - 7.26T + 1.33e3T^{2} \) |
| 17 | \( 1 - 27.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 210.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.60T + 2.97e4T^{2} \) |
| 37 | \( 1 + 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 350.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 69.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 705.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 42.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 259.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 71.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 953.T + 9.12e5T^{2} \) |
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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
−9.695256685137789508562521097421, −8.935447551836921053735920704407, −8.024623914581425926637285056143, −7.26697131428331865220752434409, −6.43399015431589815464890983859, −5.70586534498636697769096000516, −4.23555129281943112485801380252, −2.92723884222693918752705542129, −1.57379994446893208872307441228, −0.51147500189043879580022772051,
0.51147500189043879580022772051, 1.57379994446893208872307441228, 2.92723884222693918752705542129, 4.23555129281943112485801380252, 5.70586534498636697769096000516, 6.43399015431589815464890983859, 7.26697131428331865220752434409, 8.024623914581425926637285056143, 8.935447551836921053735920704407, 9.695256685137789508562521097421
