| L(s) = 1 | − 42.1·2-s − 1.38e4·3-s − 1.29e5·4-s + 3.90e5·5-s + 5.85e5·6-s + 3.78e6·7-s + 1.09e7·8-s + 6.36e7·9-s − 1.64e7·10-s + 1.01e9·11-s + 1.79e9·12-s + 5.30e8·13-s − 1.59e8·14-s − 5.42e9·15-s + 1.64e10·16-s + 3.99e8·17-s − 2.68e9·18-s − 1.38e11·19-s − 5.05e10·20-s − 5.26e10·21-s − 4.28e10·22-s + 6.06e11·23-s − 1.52e11·24-s + 1.52e11·25-s − 2.23e10·26-s + 9.08e11·27-s − 4.89e11·28-s + ⋯ |
| L(s) = 1 | − 0.116·2-s − 1.22·3-s − 0.986·4-s + 0.447·5-s + 0.142·6-s + 0.248·7-s + 0.231·8-s + 0.493·9-s − 0.0520·10-s + 1.42·11-s + 1.20·12-s + 0.180·13-s − 0.0289·14-s − 0.546·15-s + 0.959·16-s + 0.0138·17-s − 0.0574·18-s − 1.87·19-s − 0.441·20-s − 0.303·21-s − 0.166·22-s + 1.61·23-s − 0.282·24-s + 0.200·25-s − 0.0210·26-s + 0.619·27-s − 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.9154943764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9154943764\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 + 42.1T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.38e4T + 1.29e8T^{2} \) |
| 7 | \( 1 - 3.78e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.01e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 5.30e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.99e8T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.38e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.06e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 2.50e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 8.40e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.79e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 2.61e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.23e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.31e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.10e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.81e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.07e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.63e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 4.09e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 3.39e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.66e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.02e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.41e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.35e16T + 5.95e33T^{2} \) |
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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
−19.12516711354049544513579160102, −17.52760301754205548641352522724, −16.97184868396499349281845568665, −14.43850499953354706739244332423, −12.63575763431364271797657075103, −10.88429768527273124802254675646, −9.007357013575578198400944637326, −6.23251696957918084760928191908, −4.54354435880039083086726394021, −0.900878131228105644880959382096,
0.900878131228105644880959382096, 4.54354435880039083086726394021, 6.23251696957918084760928191908, 9.007357013575578198400944637326, 10.88429768527273124802254675646, 12.63575763431364271797657075103, 14.43850499953354706739244332423, 16.97184868396499349281845568665, 17.52760301754205548641352522724, 19.12516711354049544513579160102
