| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s + 7-s + 8·8-s + 9·9-s + 42·11-s + 12·12-s + 67·13-s + 2·14-s + 16·16-s − 54·17-s + 18·18-s − 115·19-s + 3·21-s + 84·22-s + 162·23-s + 24·24-s + 134·26-s + 27·27-s + 4·28-s − 210·29-s − 193·31-s + 32·32-s + 126·33-s − 108·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.0539·7-s + 0.353·8-s + 1/3·9-s + 1.15·11-s + 0.288·12-s + 1.42·13-s + 0.0381·14-s + 1/4·16-s − 0.770·17-s + 0.235·18-s − 1.38·19-s + 0.0311·21-s + 0.814·22-s + 1.46·23-s + 0.204·24-s + 1.01·26-s + 0.192·27-s + 0.0269·28-s − 1.34·29-s − 1.11·31-s + 0.176·32-s + 0.664·33-s − 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.224515766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224515766\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 67 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 115 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 193 T + p^{3} T^{2} \) |
| 37 | \( 1 - 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 12 T + p^{3} T^{2} \) |
| 43 | \( 1 + 263 T + p^{3} T^{2} \) |
| 47 | \( 1 + 414 T + p^{3} T^{2} \) |
| 53 | \( 1 - 192 T + p^{3} T^{2} \) |
| 59 | \( 1 - 690 T + p^{3} T^{2} \) |
| 61 | \( 1 + 733 T + p^{3} T^{2} \) |
| 67 | \( 1 + 299 T + p^{3} T^{2} \) |
| 71 | \( 1 + 228 T + p^{3} T^{2} \) |
| 73 | \( 1 + 938 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 462 T + p^{3} T^{2} \) |
| 89 | \( 1 + 240 T + p^{3} T^{2} \) |
| 97 | \( 1 - 511 T + p^{3} T^{2} \) |
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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
−13.02715516540034624791949670652, −11.48385477642955602521536665456, −10.85034129365183802931097398565, −9.269144123003039726175920553268, −8.479245164692666336616549071318, −7.00051324922382464871860227492, −6.08085168318161237065649004295, −4.42255779780141063873830973752, −3.42729167305411717359629086553, −1.69672953773428968538898085002,
1.69672953773428968538898085002, 3.42729167305411717359629086553, 4.42255779780141063873830973752, 6.08085168318161237065649004295, 7.00051324922382464871860227492, 8.479245164692666336616549071318, 9.269144123003039726175920553268, 10.85034129365183802931097398565, 11.48385477642955602521536665456, 13.02715516540034624791949670652
