| L(s) = 1 | + 4·3-s − 25·5-s + 192·7-s − 227·9-s + 148·11-s − 286·13-s − 100·15-s − 1.67e3·17-s − 1.06e3·19-s + 768·21-s + 2.97e3·23-s + 625·25-s − 1.88e3·27-s + 3.41e3·29-s − 2.44e3·31-s + 592·33-s − 4.80e3·35-s − 182·37-s − 1.14e3·39-s − 9.39e3·41-s + 1.24e3·43-s + 5.67e3·45-s − 1.20e4·47-s + 2.00e4·49-s − 6.71e3·51-s − 2.38e4·53-s − 3.70e3·55-s + ⋯ |
| L(s) = 1 | + 0.256·3-s − 0.447·5-s + 1.48·7-s − 0.934·9-s + 0.368·11-s − 0.469·13-s − 0.114·15-s − 1.40·17-s − 0.673·19-s + 0.380·21-s + 1.17·23-s + 1/5·25-s − 0.496·27-s + 0.752·29-s − 0.457·31-s + 0.0946·33-s − 0.662·35-s − 0.0218·37-s − 0.120·39-s − 0.873·41-s + 0.102·43-s + 0.417·45-s − 0.798·47-s + 1.19·49-s − 0.361·51-s − 1.16·53-s − 0.164·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 4 T + p^{5} T^{2} \) |
| 7 | \( 1 - 192 T + p^{5} T^{2} \) |
| 11 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 22 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1678 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2976 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3410 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2448 T + p^{5} T^{2} \) |
| 37 | \( 1 + 182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9398 T + p^{5} T^{2} \) |
| 43 | \( 1 - 1244 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12088 T + p^{5} T^{2} \) |
| 53 | \( 1 + 23846 T + p^{5} T^{2} \) |
| 59 | \( 1 - 20020 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32302 T + p^{5} T^{2} \) |
| 67 | \( 1 + 60972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 38774 T + p^{5} T^{2} \) |
| 79 | \( 1 + 33360 T + p^{5} T^{2} \) |
| 83 | \( 1 + 16716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 101370 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119038 T + p^{5} T^{2} \) |
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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
−10.61578830480697537102032526218, −9.030032925809033336983419967336, −8.527347101144338499907190709923, −7.61228413897650899581040669800, −6.50534313788166096068015292680, −5.09683710516651555382329038585, −4.34516645683517783212093349085, −2.85172142027739288391881065617, −1.64116969739870344240112165873, 0,
1.64116969739870344240112165873, 2.85172142027739288391881065617, 4.34516645683517783212093349085, 5.09683710516651555382329038585, 6.50534313788166096068015292680, 7.61228413897650899581040669800, 8.527347101144338499907190709923, 9.030032925809033336983419967336, 10.61578830480697537102032526218
