| L(s) = 1 | − 519.·2-s − 6.56e3·3-s + 1.38e5·4-s + 3.40e6·6-s − 2.20e7·7-s − 3.75e6·8-s + 4.30e7·9-s − 1.24e9·11-s − 9.07e8·12-s + 4.30e9·13-s + 1.14e10·14-s − 1.61e10·16-s − 2.30e10·17-s − 2.23e10·18-s − 4.02e9·19-s + 1.44e11·21-s + 6.44e11·22-s + 5.76e11·23-s + 2.46e10·24-s − 2.23e12·26-s − 2.82e11·27-s − 3.04e12·28-s − 4.15e12·29-s − 3.90e12·31-s + 8.88e12·32-s + 8.14e12·33-s + 1.19e13·34-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 0.577·3-s + 1.05·4-s + 0.827·6-s − 1.44·7-s − 0.0790·8-s + 0.333·9-s − 1.74·11-s − 0.609·12-s + 1.46·13-s + 2.07·14-s − 0.941·16-s − 0.800·17-s − 0.477·18-s − 0.0544·19-s + 0.833·21-s + 2.50·22-s + 1.53·23-s + 0.0456·24-s − 2.09·26-s − 0.192·27-s − 1.52·28-s − 1.54·29-s − 0.823·31-s + 1.42·32-s + 1.00·33-s + 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 519.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.20e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.24e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.30e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.30e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.02e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.76e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.15e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.90e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.59e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 2.13e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.22e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.29e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.09e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 9.51e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.05e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.73e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.20e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.67e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.95e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.10e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.12e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.37e17T + 5.95e33T^{2} \) |
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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
−10.66063905141297416963670218975, −9.535725857464917214835093936283, −8.693806539166565526948155354184, −7.43225336889438668355222607865, −6.51336614614876116884348032936, −5.29977096054588400424817946446, −3.54459055957200029647908116314, −2.18670933325314131813146850239, −0.75524061766662554354990772487, 0,
0.75524061766662554354990772487, 2.18670933325314131813146850239, 3.54459055957200029647908116314, 5.29977096054588400424817946446, 6.51336614614876116884348032936, 7.43225336889438668355222607865, 8.693806539166565526948155354184, 9.535725857464917214835093936283, 10.66063905141297416963670218975
