| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 2·10-s − 2·11-s − 2·12-s + 4·13-s + 4·14-s + 4·15-s + 16-s − 2·17-s − 18-s + 6·19-s − 2·20-s + 8·21-s + 2·22-s − 8·23-s + 2·24-s − 25-s − 4·26-s + 4·27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.577·12-s + 1.10·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s + 1.74·21-s + 0.426·22-s − 1.66·23-s + 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.56035032362304818252529920151, −12.35357978253783595070241093278, −11.53740614580497355981935805267, −10.59452070172136053574436879482, −9.489661146940125257280263860207, −8.024357451025613311326857967234, −6.70748925180204914131686110030, −5.69856539643665466176583402943, −3.54075497823589975288034605375, 0,
3.54075497823589975288034605375, 5.69856539643665466176583402943, 6.70748925180204914131686110030, 8.024357451025613311326857967234, 9.489661146940125257280263860207, 10.59452070172136053574436879482, 11.53740614580497355981935805267, 12.35357978253783595070241093278, 13.56035032362304818252529920151
