| L(s) = 1 | + 3.56·2-s − 3·3-s + 4.68·4-s − 10.6·6-s − 7·7-s − 11.8·8-s + 9·9-s − 5.19·11-s − 14.0·12-s + 54.5·13-s − 24.9·14-s − 79.5·16-s + 16.1·17-s + 32.0·18-s + 87.4·19-s + 21·21-s − 18.4·22-s + 176.·23-s + 35.4·24-s + 194.·26-s − 27·27-s − 32.7·28-s + 142.·29-s − 94.3·31-s − 188.·32-s + 15.5·33-s + 57.5·34-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.585·4-s − 0.726·6-s − 0.377·7-s − 0.521·8-s + 0.333·9-s − 0.142·11-s − 0.338·12-s + 1.16·13-s − 0.475·14-s − 1.24·16-s + 0.230·17-s + 0.419·18-s + 1.05·19-s + 0.218·21-s − 0.179·22-s + 1.59·23-s + 0.301·24-s + 1.46·26-s − 0.192·27-s − 0.221·28-s + 0.910·29-s − 0.546·31-s − 1.04·32-s + 0.0821·33-s + 0.290·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.928028697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.928028697\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.56T + 8T^{2} \) |
| 11 | \( 1 + 5.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 17.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 521.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 793.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 425.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 843.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−10.85919753065442622498907170085, −9.601977875832480089579576227904, −8.778351498589352675205054411990, −7.39551954694108854298745981874, −6.40980288269180802578694535687, −5.67587338633174292972238141469, −4.85052483572484563262538781795, −3.77827571140795938008556959803, −2.86327279723027258438942442193, −0.929427952112897476085075556694,
0.929427952112897476085075556694, 2.86327279723027258438942442193, 3.77827571140795938008556959803, 4.85052483572484563262538781795, 5.67587338633174292972238141469, 6.40980288269180802578694535687, 7.39551954694108854298745981874, 8.778351498589352675205054411990, 9.601977875832480089579576227904, 10.85919753065442622498907170085
