| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 43·11-s + 12·12-s + 18·13-s − 14·14-s + 16·16-s − 92·17-s + 18·18-s + 6·19-s − 21·21-s − 86·22-s − 129·23-s + 24·24-s + 36·26-s + 27·27-s − 28·28-s − 111·29-s − 142·31-s + 32·32-s − 129·33-s − 184·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.17·11-s + 0.288·12-s + 0.384·13-s − 0.267·14-s + 1/4·16-s − 1.31·17-s + 0.235·18-s + 0.0724·19-s − 0.218·21-s − 0.833·22-s − 1.16·23-s + 0.204·24-s + 0.271·26-s + 0.192·27-s − 0.188·28-s − 0.710·29-s − 0.822·31-s + 0.176·32-s − 0.680·33-s − 0.928·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 92 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 T + p^{3} T^{2} \) |
| 23 | \( 1 + 129 T + p^{3} T^{2} \) |
| 29 | \( 1 + 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 142 T + p^{3} T^{2} \) |
| 37 | \( 1 - 173 T + p^{3} T^{2} \) |
| 41 | \( 1 + 128 T + p^{3} T^{2} \) |
| 43 | \( 1 - 217 T + p^{3} T^{2} \) |
| 47 | \( 1 + 194 T + p^{3} T^{2} \) |
| 53 | \( 1 + 622 T + p^{3} T^{2} \) |
| 59 | \( 1 + 22 T + p^{3} T^{2} \) |
| 61 | \( 1 - 160 T + p^{3} T^{2} \) |
| 67 | \( 1 + 189 T + p^{3} T^{2} \) |
| 71 | \( 1 - 769 T + p^{3} T^{2} \) |
| 73 | \( 1 + 652 T + p^{3} T^{2} \) |
| 79 | \( 1 + 773 T + p^{3} T^{2} \) |
| 83 | \( 1 + 314 T + p^{3} T^{2} \) |
| 89 | \( 1 - 470 T + p^{3} T^{2} \) |
| 97 | \( 1 - 470 T + p^{3} T^{2} \) |
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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
−9.116722827175221832682252571725, −8.164966703334906233824896570018, −7.46508564596368178553665061643, −6.49037546744151932285596561776, −5.65966388604629321290568178498, −4.61718460598169769040393901425, −3.74052631577470048764454744399, −2.74661602335703218105156065952, −1.87038549379328327714679642887, 0,
1.87038549379328327714679642887, 2.74661602335703218105156065952, 3.74052631577470048764454744399, 4.61718460598169769040393901425, 5.65966388604629321290568178498, 6.49037546744151932285596561776, 7.46508564596368178553665061643, 8.164966703334906233824896570018, 9.116722827175221832682252571725
