| L(s) = 1 | − 2.31·2-s − 5·3-s − 2.63·4-s − 5·5-s + 11.5·6-s + 24.6·8-s − 2·9-s + 11.5·10-s + 46.2·11-s + 13.1·12-s + 61.3·13-s + 25·15-s − 36·16-s − 101.·17-s + 4.63·18-s − 3.66·19-s + 13.1·20-s − 107.·22-s + 84.8·23-s − 123.·24-s + 25·25-s − 142.·26-s + 145·27-s + 30.1·29-s − 57.9·30-s − 188.·31-s − 113.·32-s + ⋯ |
| L(s) = 1 | − 0.819·2-s − 0.962·3-s − 0.329·4-s − 0.447·5-s + 0.788·6-s + 1.08·8-s − 0.0740·9-s + 0.366·10-s + 1.26·11-s + 0.316·12-s + 1.30·13-s + 0.430·15-s − 0.562·16-s − 1.44·17-s + 0.0606·18-s − 0.0442·19-s + 0.147·20-s − 1.03·22-s + 0.769·23-s − 1.04·24-s + 0.200·25-s − 1.07·26-s + 1.03·27-s + 0.193·29-s − 0.352·30-s − 1.09·31-s − 0.627·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.31T + 8T^{2} \) |
| 3 | \( 1 + 5T + 27T^{2} \) |
| 11 | \( 1 - 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.66T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 667.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 57.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 552.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 233.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 218.T + 9.12e5T^{2} \) |
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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
−11.17640015452778181744863073085, −10.34679310526481175010457988521, −8.882492241037928788796798324180, −8.688520042377446474929939395402, −7.11204992735264282915461643907, −6.22942758318084413538326499999, −4.86943917233339096392053040151, −3.78191278706345279968320018666, −1.31984406205137736308868858378, 0,
1.31984406205137736308868858378, 3.78191278706345279968320018666, 4.86943917233339096392053040151, 6.22942758318084413538326499999, 7.11204992735264282915461643907, 8.688520042377446474929939395402, 8.882492241037928788796798324180, 10.34679310526481175010457988521, 11.17640015452778181744863073085
