| L(s) = 1 | + 0.347·3-s + 1.53·5-s + 0.305·7-s − 2.87·9-s − 0.305·11-s + 0.347·13-s + 0.532·15-s − 1.87·17-s + 0.106·21-s + 2.83·23-s − 2.65·25-s − 2.04·27-s + 3.50·29-s − 5.82·31-s − 0.106·33-s + 0.467·35-s − 8.12·37-s + 0.120·39-s + 2.47·41-s + 1.16·43-s − 4.41·45-s + 6.24·47-s − 6.90·49-s − 0.652·51-s − 7.20·53-s − 0.467·55-s + 11.3·59-s + ⋯ |
| L(s) = 1 | + 0.200·3-s + 0.685·5-s + 0.115·7-s − 0.959·9-s − 0.0920·11-s + 0.0963·13-s + 0.137·15-s − 0.455·17-s + 0.0231·21-s + 0.591·23-s − 0.530·25-s − 0.392·27-s + 0.651·29-s − 1.04·31-s − 0.0184·33-s + 0.0790·35-s − 1.33·37-s + 0.0193·39-s + 0.386·41-s + 0.177·43-s − 0.657·45-s + 0.911·47-s − 0.986·49-s − 0.0913·51-s − 0.989·53-s − 0.0630·55-s + 1.48·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 0.305T + 7T^{2} \) |
| 11 | \( 1 + 0.305T + 11T^{2} \) |
| 13 | \( 1 - 0.347T + 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 0.361T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 + 9.04T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 1.04T + 89T^{2} \) |
| 97 | \( 1 + 9.39T + 97T^{2} \) |
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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
−7.85008335134220904173596182121, −7.00698642066327844377397813168, −6.28007165953469765856533768395, −5.59241897489915924543024279731, −5.02860471725911352926052044851, −4.01456529952219604031198054256, −3.11661045025212687472974727959, −2.37188052866915527639142499354, −1.48268301837234460855463757775, 0,
1.48268301837234460855463757775, 2.37188052866915527639142499354, 3.11661045025212687472974727959, 4.01456529952219604031198054256, 5.02860471725911352926052044851, 5.59241897489915924543024279731, 6.28007165953469765856533768395, 7.00698642066327844377397813168, 7.85008335134220904173596182121
