| L(s) = 1 | − 243·3-s + 7.13e3·5-s + 1.95e4·7-s + 5.90e4·9-s − 1.96e5·11-s − 3.61e5·13-s − 1.73e6·15-s − 1.30e5·17-s + 1.85e7·19-s − 4.74e6·21-s − 2.15e7·23-s + 2.00e6·25-s − 1.43e7·27-s − 1.91e8·29-s − 2.07e8·31-s + 4.76e7·33-s + 1.39e8·35-s + 2.00e8·37-s + 8.78e7·39-s − 1.43e9·41-s + 7.12e8·43-s + 4.21e8·45-s + 4.96e8·47-s − 1.59e9·49-s + 3.18e7·51-s + 3.35e9·53-s − 1.39e9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.02·5-s + 0.439·7-s + 1/3·9-s − 0.367·11-s − 0.269·13-s − 0.589·15-s − 0.0223·17-s + 1.71·19-s − 0.253·21-s − 0.698·23-s + 0.0411·25-s − 0.192·27-s − 1.73·29-s − 1.30·31-s + 0.212·33-s + 0.448·35-s + 0.476·37-s + 0.155·39-s − 1.93·41-s + 0.739·43-s + 0.340·45-s + 0.315·47-s − 0.806·49-s + 0.0129·51-s + 1.10·53-s − 0.374·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{5} T \) |
| good | 5 | \( 1 - 1426 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 19536 T + p^{11} T^{2} \) |
| 11 | \( 1 + 196148 T + p^{11} T^{2} \) |
| 13 | \( 1 + 361414 T + p^{11} T^{2} \) |
| 17 | \( 1 + 130942 T + p^{11} T^{2} \) |
| 19 | \( 1 - 18516692 T + p^{11} T^{2} \) |
| 23 | \( 1 + 21560872 T + p^{11} T^{2} \) |
| 29 | \( 1 + 191663742 T + p^{11} T^{2} \) |
| 31 | \( 1 + 207933800 T + p^{11} T^{2} \) |
| 37 | \( 1 - 200784930 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1435256598 T + p^{11} T^{2} \) |
| 43 | \( 1 - 712703116 T + p^{11} T^{2} \) |
| 47 | \( 1 - 496082400 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3350114330 T + p^{11} T^{2} \) |
| 59 | \( 1 - 4583222956 T + p^{11} T^{2} \) |
| 61 | \( 1 + 3427501702 T + p^{11} T^{2} \) |
| 67 | \( 1 - 17079378356 T + p^{11} T^{2} \) |
| 71 | \( 1 - 7915078504 T + p^{11} T^{2} \) |
| 73 | \( 1 - 31559658778 T + p^{11} T^{2} \) |
| 79 | \( 1 + 41023578808 T + p^{11} T^{2} \) |
| 83 | \( 1 + 19974672172 T + p^{11} T^{2} \) |
| 89 | \( 1 + 10640163606 T + p^{11} T^{2} \) |
| 97 | \( 1 - 6441105794 T + p^{11} T^{2} \) |
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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
−9.961870704809884894239759400071, −9.353695125931287970799116313253, −7.907988202836601718264491820334, −6.95067026874789905557396939535, −5.60711551489054240412306227122, −5.26992365758035423254343712331, −3.73047116420681354403655276027, −2.24119437443110786223699901613, −1.34708840012423364950837910538, 0,
1.34708840012423364950837910538, 2.24119437443110786223699901613, 3.73047116420681354403655276027, 5.26992365758035423254343712331, 5.60711551489054240412306227122, 6.95067026874789905557396939535, 7.907988202836601718264491820334, 9.353695125931287970799116313253, 9.961870704809884894239759400071
