| L(s) = 1 | − 233.·3-s − 356.·5-s − 2.40e3·7-s + 3.46e4·9-s − 7.28e4·11-s + 3.93e4·13-s + 8.31e4·15-s − 5.11e5·17-s − 9.00e5·19-s + 5.59e5·21-s − 3.82e5·23-s − 1.82e6·25-s − 3.48e6·27-s − 4.73e6·29-s + 7.93e5·31-s + 1.69e7·33-s + 8.56e5·35-s − 1.72e7·37-s − 9.16e6·39-s − 1.53e7·41-s − 1.87e7·43-s − 1.23e7·45-s + 4.68e7·47-s + 5.76e6·49-s + 1.19e8·51-s + 2.06e7·53-s + 2.60e7·55-s + ⋯ |
| L(s) = 1 | − 1.66·3-s − 0.255·5-s − 0.377·7-s + 1.75·9-s − 1.50·11-s + 0.382·13-s + 0.424·15-s − 1.48·17-s − 1.58·19-s + 0.627·21-s − 0.285·23-s − 0.934·25-s − 1.26·27-s − 1.24·29-s + 0.154·31-s + 2.49·33-s + 0.0964·35-s − 1.51·37-s − 0.634·39-s − 0.848·41-s − 0.834·43-s − 0.449·45-s + 1.40·47-s + 0.142·49-s + 2.46·51-s + 0.360·53-s + 0.383·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.04850010179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04850010179\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 + 233.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 356.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 7.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.00e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.82e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.93e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.72e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.00e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.69e8T + 7.60e17T^{2} \) |
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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
−11.70821549297374382700664845082, −10.81869814273198786265970996829, −10.23143695870121057083905311385, −8.578989212147553950771210624452, −7.14594433182344877090832915868, −6.16318604668279791165474188114, −5.20699523842490408126911721651, −4.05724606289318206373335108054, −2.06474294266778208537135673935, −0.12250774552239681175279965197,
0.12250774552239681175279965197, 2.06474294266778208537135673935, 4.05724606289318206373335108054, 5.20699523842490408126911721651, 6.16318604668279791165474188114, 7.14594433182344877090832915868, 8.578989212147553950771210624452, 10.23143695870121057083905311385, 10.81869814273198786265970996829, 11.70821549297374382700664845082
