| L(s) = 1 | − 228·3-s + 666·5-s − 6.32e3·7-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 + 1.44e6·21-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 − 4.21e6·35-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 − 3.10e5·49-s − 1.34e8·51-s + 6.63e7·53-s + 2.02e7·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.476·5-s − 0.996·7-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 + 1.61·21-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 − 0.474·35-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 − 0.00768·49-s − 2.78·51-s + 1.15·53-s + 0.298·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 76 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 666 T + p^{9} T^{2} \) |
| 7 | \( 1 + 904 p T + p^{9} T^{2} \) |
| 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
−12.36319694935754290292366250001, −11.38074622469042908518724752416, −10.26795581413172997394865430083, −9.358266762540136142301887226590, −7.23555106257850337532494062732, −6.10930472903503615612578376456, −5.39596505765280699147347103376, −3.62462091406429405752104232391, −1.33899012374413861333781119107, 0,
1.33899012374413861333781119107, 3.62462091406429405752104232391, 5.39596505765280699147347103376, 6.10930472903503615612578376456, 7.23555106257850337532494062732, 9.358266762540136142301887226590, 10.26795581413172997394865430083, 11.38074622469042908518724752416, 12.36319694935754290292366250001
