| L(s) = 1 | − 29.7·2-s − 81·3-s + 373.·4-s + 2.40e3·6-s + 1.07e4·7-s + 4.13e3·8-s + 6.56e3·9-s + 5.44e4·11-s − 3.02e4·12-s + 5.35e4·13-s − 3.18e5·14-s − 3.13e5·16-s + 6.44e5·17-s − 1.95e5·18-s − 2.10e5·19-s − 8.67e5·21-s − 1.61e6·22-s + 9.52e4·23-s − 3.34e5·24-s − 1.59e6·26-s − 5.31e5·27-s + 3.99e6·28-s − 2.25e6·29-s − 5.48e5·31-s + 7.22e6·32-s − 4.40e6·33-s − 1.91e7·34-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.728·4-s + 0.759·6-s + 1.68·7-s + 0.356·8-s + 0.333·9-s + 1.12·11-s − 0.420·12-s + 0.519·13-s − 2.21·14-s − 1.19·16-s + 1.87·17-s − 0.438·18-s − 0.370·19-s − 0.973·21-s − 1.47·22-s + 0.0709·23-s − 0.205·24-s − 0.683·26-s − 0.192·27-s + 1.22·28-s − 0.591·29-s − 0.106·31-s + 1.21·32-s − 0.647·33-s − 2.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.204532272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204532272\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 29.7T + 512T^{2} \) |
| 7 | \( 1 - 1.07e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.35e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.10e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.52e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.48e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.19e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.48e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.80e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.85e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.05e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.84e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.07e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.46e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.06e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.31e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.40e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.04e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.28e9T + 7.60e17T^{2} \) |
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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
−12.16532408645665190541402672165, −11.26932832919934287514199108512, −10.46105862784767973128261256799, −9.200373287082590416489885943406, −8.172485543775915776479629521127, −7.25229706006745654740423097978, −5.59294248852461961521132350431, −4.19236877393513803169972526732, −1.63723725915114637656183449414, −0.934411450930870695793487568486,
0.934411450930870695793487568486, 1.63723725915114637656183449414, 4.19236877393513803169972526732, 5.59294248852461961521132350431, 7.25229706006745654740423097978, 8.172485543775915776479629521127, 9.200373287082590416489885943406, 10.46105862784767973128261256799, 11.26932832919934287514199108512, 12.16532408645665190541402672165
