| L(s) = 1 | − 2·2-s + 4·3-s − 28·4-s − 8·6-s − 192·7-s + 120·8-s − 227·9-s − 148·11-s − 112·12-s − 286·13-s + 384·14-s + 656·16-s + 1.67e3·17-s + 454·18-s + 1.06e3·19-s − 768·21-s + 296·22-s − 2.97e3·23-s + 480·24-s + 572·26-s − 1.88e3·27-s + 5.37e3·28-s − 3.41e3·29-s − 2.44e3·31-s − 5.15e3·32-s − 592·33-s − 3.35e3·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.256·3-s − 7/8·4-s − 0.0907·6-s − 1.48·7-s + 0.662·8-s − 0.934·9-s − 0.368·11-s − 0.224·12-s − 0.469·13-s + 0.523·14-s + 0.640·16-s + 1.40·17-s + 0.330·18-s + 0.673·19-s − 0.380·21-s + 0.130·22-s − 1.17·23-s + 0.170·24-s + 0.165·26-s − 0.496·27-s + 1.29·28-s − 0.752·29-s − 0.457·31-s − 0.889·32-s − 0.0946·33-s − 0.497·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 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
−16.17828104866988336448239195244, −14.48727191303206209106814016356, −13.44559404276664901662399495488, −12.22674507089581931954142321814, −10.14043978508760786305504670386, −9.255424774101427276845588504649, −7.76787632260424582187469712466, −5.65541904075737917698205995072, −3.36605588359824232335034067530, 0,
3.36605588359824232335034067530, 5.65541904075737917698205995072, 7.76787632260424582187469712466, 9.255424774101427276845588504649, 10.14043978508760786305504670386, 12.22674507089581931954142321814, 13.44559404276664901662399495488, 14.48727191303206209106814016356, 16.17828104866988336448239195244
