| L(s) = 1 | − 16·3-s + 16·5-s + 13·9-s − 76·11-s + 880·13-s − 256·15-s − 1.05e3·17-s + 1.93e3·19-s + 936·23-s − 2.86e3·25-s + 3.68e3·27-s − 3.98e3·29-s + 1.56e3·31-s + 1.21e3·33-s + 4.93e3·37-s − 1.40e4·39-s − 1.58e4·41-s − 1.64e4·43-s + 208·45-s − 2.07e4·47-s + 1.68e4·51-s − 3.74e4·53-s − 1.21e3·55-s − 3.09e4·57-s + 2.11e4·59-s − 2.99e3·61-s + 1.40e4·65-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 0.286·5-s + 0.0534·9-s − 0.189·11-s + 1.44·13-s − 0.293·15-s − 0.886·17-s + 1.23·19-s + 0.368·23-s − 0.918·25-s + 0.971·27-s − 0.879·29-s + 0.293·31-s + 0.194·33-s + 0.592·37-s − 1.48·39-s − 1.47·41-s − 1.35·43-s + 0.0153·45-s − 1.37·47-s + 0.909·51-s − 1.82·53-s − 0.0542·55-s − 1.26·57-s + 0.790·59-s − 0.102·61-s + 0.413·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 16 T + p^{5} T^{2} \) |
| 5 | \( 1 - 16 T + p^{5} T^{2} \) |
| 11 | \( 1 + 76 T + p^{5} T^{2} \) |
| 13 | \( 1 - 880 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1056 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1936 T + p^{5} T^{2} \) |
| 23 | \( 1 - 936 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3982 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1568 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4938 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15840 T + p^{5} T^{2} \) |
| 43 | \( 1 + 16412 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20768 T + p^{5} T^{2} \) |
| 53 | \( 1 + 37402 T + p^{5} T^{2} \) |
| 59 | \( 1 - 21136 T + p^{5} T^{2} \) |
| 61 | \( 1 + 2992 T + p^{5} T^{2} \) |
| 67 | \( 1 + 45836 T + p^{5} T^{2} \) |
| 71 | \( 1 + 49840 T + p^{5} T^{2} \) |
| 73 | \( 1 + 56320 T + p^{5} T^{2} \) |
| 79 | \( 1 - 40744 T + p^{5} T^{2} \) |
| 83 | \( 1 - 112464 T + p^{5} T^{2} \) |
| 89 | \( 1 - 64256 T + p^{5} T^{2} \) |
| 97 | \( 1 + 2272 T + p^{5} T^{2} \) |
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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
−11.30993833705951049102484630701, −10.33827622522800303758518587013, −9.202464916592899095833117710846, −8.079338207150592065476131412267, −6.65820021598812265171019338656, −5.86798867726540691966244533113, −4.88215768015196385897094500276, −3.33606421234861251947254713755, −1.48732122041019195017557348087, 0,
1.48732122041019195017557348087, 3.33606421234861251947254713755, 4.88215768015196385897094500276, 5.86798867726540691966244533113, 6.65820021598812265171019338656, 8.079338207150592065476131412267, 9.202464916592899095833117710846, 10.33827622522800303758518587013, 11.30993833705951049102484630701
