| L(s) = 1 | − 345.·2-s + 1.26e4·3-s − 1.16e4·4-s − 1.32e6·5-s − 4.38e6·6-s + 1.45e6·7-s + 4.93e7·8-s + 3.21e7·9-s + 4.59e8·10-s − 2.01e8·11-s − 1.47e8·12-s − 1.88e9·13-s − 5.04e8·14-s − 1.68e10·15-s − 1.55e10·16-s − 3.25e9·17-s − 1.11e10·18-s + 1.31e11·19-s + 1.54e10·20-s + 1.85e10·21-s + 6.95e10·22-s + 3.00e11·23-s + 6.26e11·24-s + 1.00e12·25-s + 6.51e11·26-s − 1.23e12·27-s − 1.69e10·28-s + ⋯ |
| L(s) = 1 | − 0.954·2-s + 1.11·3-s − 0.0887·4-s − 1.52·5-s − 1.06·6-s + 0.0956·7-s + 1.03·8-s + 0.248·9-s + 1.45·10-s − 0.282·11-s − 0.0991·12-s − 0.641·13-s − 0.0913·14-s − 1.70·15-s − 0.903·16-s − 0.113·17-s − 0.237·18-s + 1.77·19-s + 0.135·20-s + 0.106·21-s + 0.270·22-s + 0.799·23-s + 1.16·24-s + 1.31·25-s + 0.612·26-s − 0.839·27-s − 0.00849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 59 | \( 1 + 1.46e14T \) |
| good | 2 | \( 1 + 345.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.26e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 1.32e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.45e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 2.01e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.88e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.25e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.31e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 3.00e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.76e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 2.46e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 5.69e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.83e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.95e12T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.27e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.13e13T + 2.05e29T^{2} \) |
| 61 | \( 1 + 9.96e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.02e13T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.76e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 4.52e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.36e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.39e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.40e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.34e16T + 5.95e33T^{2} \) |
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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
−10.95552263011801105251764873157, −9.590955304313608017964013593804, −8.770027168590637801981558142940, −7.76080741656909005584794219921, −7.42827301292106048102548436024, −4.92225399985105093490019470151, −3.75015717951039908304615415171, −2.70109770193988564418650260308, −1.06379183501010261877304444890, 0,
1.06379183501010261877304444890, 2.70109770193988564418650260308, 3.75015717951039908304615415171, 4.92225399985105093490019470151, 7.42827301292106048102548436024, 7.76080741656909005584794219921, 8.770027168590637801981558142940, 9.590955304313608017964013593804, 10.95552263011801105251764873157
