| L(s) = 1 | + 3·3-s − 24·7-s + 9·9-s + 28·11-s + 74·13-s − 82·17-s − 92·19-s − 72·21-s + 8·23-s + 27·27-s − 138·29-s − 80·31-s + 84·33-s − 30·37-s + 222·39-s + 282·41-s + 4·43-s + 240·47-s + 233·49-s − 246·51-s + 130·53-s − 276·57-s − 596·59-s − 218·61-s − 216·63-s − 436·67-s + 24·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.29·7-s + 1/3·9-s + 0.767·11-s + 1.57·13-s − 1.16·17-s − 1.11·19-s − 0.748·21-s + 0.0725·23-s + 0.192·27-s − 0.883·29-s − 0.463·31-s + 0.443·33-s − 0.133·37-s + 0.911·39-s + 1.07·41-s + 0.0141·43-s + 0.744·47-s + 0.679·49-s − 0.675·51-s + 0.336·53-s − 0.641·57-s − 1.31·59-s − 0.457·61-s − 0.431·63-s − 0.795·67-s + 0.0418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 82 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 T + p^{3} T^{2} \) |
| 29 | \( 1 + 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 130 T + p^{3} T^{2} \) |
| 59 | \( 1 + 596 T + p^{3} T^{2} \) |
| 61 | \( 1 + 218 T + p^{3} T^{2} \) |
| 67 | \( 1 + 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 856 T + p^{3} T^{2} \) |
| 73 | \( 1 - 998 T + p^{3} T^{2} \) |
| 79 | \( 1 - 32 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 + 246 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−9.041093257673981267548946331721, −8.391780627434546801330143803742, −7.19311183474821832049663019585, −6.43892228991003697708515777172, −5.89724835141326367530640501919, −4.23988070984666888245190236853, −3.73362965828773138598976804996, −2.68973481271773316563340366736, −1.48234592030163062836910378067, 0,
1.48234592030163062836910378067, 2.68973481271773316563340366736, 3.73362965828773138598976804996, 4.23988070984666888245190236853, 5.89724835141326367530640501919, 6.43892228991003697708515777172, 7.19311183474821832049663019585, 8.391780627434546801330143803742, 9.041093257673981267548946331721
