| L(s) = 1 | − 1.20e4·3-s − 3.90e5·5-s + 9.53e6·7-s + 1.57e7·9-s − 4.01e8·11-s + 8.56e8·13-s + 4.70e9·15-s − 3.89e10·17-s + 1.13e11·19-s − 1.14e11·21-s − 1.64e10·23-s + 1.52e11·25-s + 1.36e12·27-s − 2.27e12·29-s + 1.63e12·31-s + 4.82e12·33-s − 3.72e12·35-s − 1.75e13·37-s − 1.03e13·39-s − 2.95e13·41-s − 1.37e14·43-s − 6.15e12·45-s + 1.65e14·47-s − 1.41e14·49-s + 4.68e14·51-s − 7.25e14·53-s + 1.56e14·55-s + ⋯ |
| L(s) = 1 | − 1.05·3-s − 0.447·5-s + 0.625·7-s + 0.122·9-s − 0.564·11-s + 0.291·13-s + 0.473·15-s − 1.35·17-s + 1.53·19-s − 0.662·21-s − 0.0438·23-s + 0.200·25-s + 0.929·27-s − 0.844·29-s + 0.344·31-s + 0.597·33-s − 0.279·35-s − 0.823·37-s − 0.308·39-s − 0.578·41-s − 1.79·43-s − 0.0546·45-s + 1.01·47-s − 0.609·49-s + 1.43·51-s − 1.60·53-s + 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.7586603102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7586603102\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 3 | \( 1 + 1.20e4T + 1.29e8T^{2} \) |
| 7 | \( 1 - 9.53e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.01e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 8.56e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.89e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.13e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 1.64e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.27e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 1.63e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.75e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.95e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.37e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.65e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 7.25e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.62e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.46e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 2.03e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 9.39e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.54e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.30e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 6.14e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.67e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.01e17T + 5.95e33T^{2} \) |
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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
−11.32451679594704091251989270009, −10.32758217860040952422896966976, −8.855283576591686332962823211031, −7.73351257984406849825474046457, −6.59727669109718406572507956808, −5.39936322459760723009041739800, −4.66015521605310969812591159419, −3.19870964724206091573500130032, −1.67268394953906659331437975079, −0.41295800142498264811304260845,
0.41295800142498264811304260845, 1.67268394953906659331437975079, 3.19870964724206091573500130032, 4.66015521605310969812591159419, 5.39936322459760723009041739800, 6.59727669109718406572507956808, 7.73351257984406849825474046457, 8.855283576591686332962823211031, 10.32758217860040952422896966976, 11.32451679594704091251989270009
