| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 6·11-s − 2·13-s − 14-s − 16-s − 4·17-s − 6·19-s + 6·22-s − 2·26-s + 28-s + 2·29-s − 10·31-s + 5·32-s − 4·34-s − 4·37-s − 6·38-s − 2·41-s − 4·43-s − 6·44-s + 49-s + 2·52-s − 6·53-s + 3·56-s + 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.80·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.37·19-s + 1.27·22-s − 0.392·26-s + 0.188·28-s + 0.371·29-s − 1.79·31-s + 0.883·32-s − 0.685·34-s − 0.657·37-s − 0.973·38-s − 0.312·41-s − 0.609·43-s − 0.904·44-s + 1/7·49-s + 0.277·52-s − 0.824·53-s + 0.400·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.971552311057573237641723762458, −8.564726035632039332608333026566, −7.11999628519012507821966096100, −6.49227489268027016194575261734, −5.75408159665580195145733679557, −4.60578903619236206845761227834, −4.08242632050905699099504942411, −3.20269986936965107647955994590, −1.82947662480420584992219247222, 0,
1.82947662480420584992219247222, 3.20269986936965107647955994590, 4.08242632050905699099504942411, 4.60578903619236206845761227834, 5.75408159665580195145733679557, 6.49227489268027016194575261734, 7.11999628519012507821966096100, 8.564726035632039332608333026566, 8.971552311057573237641723762458
