L(s) = 1 | + 3·3-s − 2·5-s + 9·9-s + 8·11-s + 42·13-s − 6·15-s + 2·17-s − 124·19-s − 76·23-s − 121·25-s + 27·27-s + 254·29-s − 72·31-s + 24·33-s + 398·37-s + 126·39-s − 462·41-s − 212·43-s − 18·45-s − 264·47-s + 6·51-s − 162·53-s − 16·55-s − 372·57-s − 772·59-s − 30·61-s − 84·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.178·5-s + 1/3·9-s + 0.219·11-s + 0.896·13-s − 0.103·15-s + 0.0285·17-s − 1.49·19-s − 0.689·23-s − 0.967·25-s + 0.192·27-s + 1.62·29-s − 0.417·31-s + 0.126·33-s + 1.76·37-s + 0.517·39-s − 1.75·41-s − 0.751·43-s − 0.0596·45-s − 0.819·47-s + 0.0164·51-s − 0.419·53-s − 0.0392·55-s − 0.864·57-s − 1.70·59-s − 0.0629·61-s − 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 254 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 462 T + p^{3} T^{2} \) |
| 43 | \( 1 + 212 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 236 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1190 T + p^{3} T^{2} \) |
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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
−8.283911975509419999696744443876, −7.72562921719305832425587162632, −6.49660954313214490997619646062, −6.22486127452749902796526162495, −4.89086287721154665741366060451, −4.10041059102492019766398808854, −3.37579479618716201888604923402, −2.29616018851190485992249993546, −1.37301662063749368981732339512, 0,
1.37301662063749368981732339512, 2.29616018851190485992249993546, 3.37579479618716201888604923402, 4.10041059102492019766398808854, 4.89086287721154665741366060451, 6.22486127452749902796526162495, 6.49660954313214490997619646062, 7.72562921719305832425587162632, 8.283911975509419999696744443876