| L(s) = 1 | + 5-s − 5·11-s + 4·13-s − 4·17-s + 5·19-s − 6·23-s + 25-s − 5·29-s + 9·31-s − 10·37-s + 7·41-s + 2·43-s − 2·47-s − 7·49-s + 8·53-s − 5·55-s + 59-s − 2·61-s + 4·65-s − 6·67-s − 71-s − 8·73-s − 12·79-s − 6·83-s − 4·85-s − 9·89-s + 5·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.50·11-s + 1.10·13-s − 0.970·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s − 0.928·29-s + 1.61·31-s − 1.64·37-s + 1.09·41-s + 0.304·43-s − 0.291·47-s − 49-s + 1.09·53-s − 0.674·55-s + 0.130·59-s − 0.256·61-s + 0.496·65-s − 0.733·67-s − 0.118·71-s − 0.936·73-s − 1.35·79-s − 0.658·83-s − 0.433·85-s − 0.953·89-s + 0.512·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.70715175882258329308009960961, −6.99835950622058167840985689016, −6.08438344147867886231929340267, −5.64407299768507025484510862681, −4.86154103115382654376507449448, −4.02492507772792149421985969698, −3.08606864493245058459527762378, −2.34230964433500185499999339359, −1.37354983862931044783723956819, 0,
1.37354983862931044783723956819, 2.34230964433500185499999339359, 3.08606864493245058459527762378, 4.02492507772792149421985969698, 4.86154103115382654376507449448, 5.64407299768507025484510862681, 6.08438344147867886231929340267, 6.99835950622058167840985689016, 7.70715175882258329308009960961
