| L(s) = 1 | + 2·2-s − 4·3-s − 28·4-s + 25·5-s − 8·6-s + 192·7-s − 120·8-s − 227·9-s + 50·10-s − 148·11-s + 112·12-s + 286·13-s + 384·14-s − 100·15-s + 656·16-s − 1.67e3·17-s − 454·18-s + 1.06e3·19-s − 700·20-s − 768·21-s − 296·22-s + 2.97e3·23-s + 480·24-s + 625·25-s + 572·26-s + 1.88e3·27-s − 5.37e3·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.256·3-s − 7/8·4-s + 0.447·5-s − 0.0907·6-s + 1.48·7-s − 0.662·8-s − 0.934·9-s + 0.158·10-s − 0.368·11-s + 0.224·12-s + 0.469·13-s + 0.523·14-s − 0.114·15-s + 0.640·16-s − 1.40·17-s − 0.330·18-s + 0.673·19-s − 0.391·20-s − 0.380·21-s − 0.130·22-s + 1.17·23-s + 0.170·24-s + 1/5·25-s + 0.165·26-s + 0.496·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9710655448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9710655448\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 3 | \( 1 + 4 T + p^{5} T^{2} \) |
| 7 | \( 1 - 192 T + p^{5} T^{2} \) |
| 11 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 22 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1678 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2976 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3410 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2448 T + p^{5} T^{2} \) |
| 37 | \( 1 - 182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9398 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1244 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12088 T + p^{5} T^{2} \) |
| 53 | \( 1 - 23846 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20020 T + p^{5} T^{2} \) |
| 61 | \( 1 - 32302 T + p^{5} T^{2} \) |
| 67 | \( 1 - 60972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 38774 T + p^{5} T^{2} \) |
| 79 | \( 1 + 33360 T + p^{5} T^{2} \) |
| 83 | \( 1 - 16716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 101370 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119038 T + p^{5} T^{2} \) |
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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
−23.24609425663826686791273389532, −21.93222733105611867420931259922, −20.58783903836707634657406003956, −18.22860474114662304511675025441, −17.33405168388793405888238876888, −14.76134755315793297199518363219, −13.45166559410003997536070100498, −11.25503050513697713183112168572, −8.690118924564440058768821531438, −5.18869999117144960219199272171,
5.18869999117144960219199272171, 8.690118924564440058768821531438, 11.25503050513697713183112168572, 13.45166559410003997536070100498, 14.76134755315793297199518363219, 17.33405168388793405888238876888, 18.22860474114662304511675025441, 20.58783903836707634657406003956, 21.93222733105611867420931259922, 23.24609425663826686791273389532
