| L(s) = 1 | − 228·3-s + 666·5-s + 3.23e4·9-s − 3.04e4·11-s + 3.23e4·13-s − 1.51e5·15-s − 5.90e5·17-s − 3.46e4·19-s + 1.04e6·23-s − 1.50e6·25-s − 2.87e6·27-s + 4.40e6·29-s + 7.40e6·31-s + 6.93e6·33-s + 1.02e7·37-s − 7.37e6·39-s − 1.83e7·41-s − 2.52e5·43-s + 2.15e7·45-s + 4.95e7·47-s + 1.34e8·51-s − 6.63e7·53-s − 2.02e7·55-s + 7.90e6·57-s + 6.15e7·59-s − 3.56e7·61-s + 2.15e7·65-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.476·5-s + 1.64·9-s − 0.626·11-s + 0.314·13-s − 0.774·15-s − 1.71·17-s − 0.0610·19-s + 0.781·23-s − 0.772·25-s − 1.04·27-s + 1.15·29-s + 1.43·31-s + 1.01·33-s + 0.897·37-s − 0.510·39-s − 1.01·41-s − 0.0112·43-s + 0.782·45-s + 1.48·47-s + 2.78·51-s − 1.15·53-s − 0.298·55-s + 0.0992·57-s + 0.661·59-s − 0.329·61-s + 0.149·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 76 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 666 T + p^{9} T^{2} \) |
| 11 | \( 1 + 30420 T + p^{9} T^{2} \) |
| 13 | \( 1 - 32338 T + p^{9} T^{2} \) |
| 17 | \( 1 + 590994 T + p^{9} T^{2} \) |
| 19 | \( 1 + 34676 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1048536 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4409406 T + p^{9} T^{2} \) |
| 31 | \( 1 - 7401184 T + p^{9} T^{2} \) |
| 37 | \( 1 - 10234502 T + p^{9} T^{2} \) |
| 41 | \( 1 + 18352746 T + p^{9} T^{2} \) |
| 43 | \( 1 + 252340 T + p^{9} T^{2} \) |
| 47 | \( 1 - 49517136 T + p^{9} T^{2} \) |
| 53 | \( 1 + 66396906 T + p^{9} T^{2} \) |
| 59 | \( 1 - 61523748 T + p^{9} T^{2} \) |
| 61 | \( 1 + 35638622 T + p^{9} T^{2} \) |
| 67 | \( 1 - 181742372 T + p^{9} T^{2} \) |
| 71 | \( 1 - 90904968 T + p^{9} T^{2} \) |
| 73 | \( 1 - 262978678 T + p^{9} T^{2} \) |
| 79 | \( 1 + 116502832 T + p^{9} T^{2} \) |
| 83 | \( 1 - 9563724 T + p^{9} T^{2} \) |
| 89 | \( 1 + 611826714 T + p^{9} T^{2} \) |
| 97 | \( 1 - 259312798 T + p^{9} T^{2} \) |
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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
−10.60811534304457445170870552413, −9.654050498603918184754718081269, −8.326259270099930529617044588736, −6.83859855512998320601489009529, −6.21437987122291653190589813289, −5.18866267650685211540886931758, −4.35453238273260656558586272865, −2.46960567930377211185531668438, −1.05344347001895307565235723757, 0,
1.05344347001895307565235723757, 2.46960567930377211185531668438, 4.35453238273260656558586272865, 5.18866267650685211540886931758, 6.21437987122291653190589813289, 6.83859855512998320601489009529, 8.326259270099930529617044588736, 9.654050498603918184754718081269, 10.60811534304457445170870552413
