| L(s) = 1 | − 464.·2-s − 6.56e3·3-s + 8.50e4·4-s + 3.05e6·6-s + 1.02e7·7-s + 2.13e7·8-s + 4.30e7·9-s + 1.01e9·11-s − 5.57e8·12-s − 1.48e9·13-s − 4.74e9·14-s − 2.10e10·16-s + 4.50e10·17-s − 2.00e10·18-s − 6.17e10·19-s − 6.69e10·21-s − 4.74e11·22-s − 2.99e11·23-s − 1.40e11·24-s + 6.90e11·26-s − 2.82e11·27-s + 8.68e11·28-s + 6.88e11·29-s − 3.25e12·31-s + 7.00e12·32-s − 6.69e12·33-s − 2.09e13·34-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.648·4-s + 0.741·6-s + 0.669·7-s + 0.450·8-s + 0.333·9-s + 1.43·11-s − 0.374·12-s − 0.504·13-s − 0.859·14-s − 1.22·16-s + 1.56·17-s − 0.428·18-s − 0.833·19-s − 0.386·21-s − 1.84·22-s − 0.796·23-s − 0.260·24-s + 0.648·26-s − 0.192·27-s + 0.434·28-s + 0.255·29-s − 0.686·31-s + 1.12·32-s − 0.828·33-s − 2.01·34-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 | \( 1 + 464.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 1.02e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.01e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.48e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 4.50e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 6.17e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.99e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 6.88e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.25e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.35e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 8.21e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 4.83e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 8.30e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.23e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 8.67e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.67e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.09e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.65e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.01e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.43e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 7.76e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.44e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.13e16T + 5.95e33T^{2} \) |
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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
−10.44211242490939369797722282778, −9.619809713234834247253309476157, −8.538752202623523335865462528472, −7.56431246273098176436880186945, −6.48378009912579564159881077715, −5.07378904479771619833332187688, −3.85883685631642764912576328528, −1.85943390562551480472022295542, −1.12456351173920841056689268050, 0,
1.12456351173920841056689268050, 1.85943390562551480472022295542, 3.85883685631642764912576328528, 5.07378904479771619833332187688, 6.48378009912579564159881077715, 7.56431246273098176436880186945, 8.538752202623523335865462528472, 9.619809713234834247253309476157, 10.44211242490939369797722282778
