| L(s) = 1 | + 1.33·2-s − 6.20·4-s + 22.3·7-s − 19.0·8-s + 28.6·11-s − 75.7·13-s + 29.9·14-s + 24.1·16-s + 126.·17-s − 72.6·19-s + 38.3·22-s − 124.·23-s − 101.·26-s − 138.·28-s − 83.6·29-s + 37.8·31-s + 184.·32-s + 169.·34-s − 256.·37-s − 97.3·38-s + 84.6·41-s + 32.9·43-s − 177.·44-s − 166.·46-s − 62.4·47-s + 155.·49-s + 469.·52-s + ⋯ |
| L(s) = 1 | + 0.473·2-s − 0.775·4-s + 1.20·7-s − 0.841·8-s + 0.784·11-s − 1.61·13-s + 0.571·14-s + 0.377·16-s + 1.80·17-s − 0.877·19-s + 0.371·22-s − 1.12·23-s − 0.765·26-s − 0.935·28-s − 0.535·29-s + 0.219·31-s + 1.01·32-s + 0.853·34-s − 1.13·37-s − 0.415·38-s + 0.322·41-s + 0.116·43-s − 0.608·44-s − 0.532·46-s − 0.193·47-s + 0.453·49-s + 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.33T + 8T^{2} \) |
| 7 | \( 1 - 22.3T + 343T^{2} \) |
| 11 | \( 1 - 28.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 83.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 84.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 62.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 493.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 187.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 930.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 821.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 833.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 537.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 106.T + 9.12e5T^{2} \) |
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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
−8.255730453468944744034697359588, −7.81652584739429076313252622756, −6.82548517467352134614900585401, −5.64962931089035898009134195765, −5.17643801561753254875375374425, −4.33132993296241551512912738337, −3.66515892718740632516173263476, −2.39708956736082055102104184262, −1.28948315761324558777951470872, 0,
1.28948315761324558777951470872, 2.39708956736082055102104184262, 3.66515892718740632516173263476, 4.33132993296241551512912738337, 5.17643801561753254875375374425, 5.64962931089035898009134195765, 6.82548517467352134614900585401, 7.81652584739429076313252622756, 8.255730453468944744034697359588
