| L(s) = 1 | + 5.21·2-s − 60.7·3-s − 100.·4-s + 125·5-s − 316.·6-s + 343·7-s − 1.19e3·8-s + 1.50e3·9-s + 651.·10-s + 7.40e3·11-s + 6.12e3·12-s + 3.78e3·13-s + 1.78e3·14-s − 7.59e3·15-s + 6.68e3·16-s + 1.92e4·17-s + 7.83e3·18-s − 1.66e4·19-s − 1.26e4·20-s − 2.08e4·21-s + 3.85e4·22-s − 3.19e4·23-s + 7.24e4·24-s + 1.56e4·25-s + 1.97e4·26-s + 4.15e4·27-s − 3.45e4·28-s + ⋯ |
| L(s) = 1 | + 0.460·2-s − 1.29·3-s − 0.787·4-s + 0.447·5-s − 0.598·6-s + 0.377·7-s − 0.823·8-s + 0.687·9-s + 0.206·10-s + 1.67·11-s + 1.02·12-s + 0.478·13-s + 0.174·14-s − 0.580·15-s + 0.408·16-s + 0.949·17-s + 0.316·18-s − 0.556·19-s − 0.352·20-s − 0.490·21-s + 0.772·22-s − 0.548·23-s + 1.06·24-s + 0.199·25-s + 0.220·26-s + 0.406·27-s − 0.297·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.326405732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.326405732\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 - 5.21T + 128T^{2} \) |
| 3 | \( 1 + 60.7T + 2.18e3T^{2} \) |
| 11 | \( 1 - 7.40e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.78e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.53e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.24e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.34e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.92e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.92e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.88e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.18e6T + 8.07e13T^{2} \) |
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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
−14.72417160241386605255424390681, −13.83719169395485278432700734985, −12.39443188719093491279321934128, −11.60289090716383510105410473431, −10.11740635326102555321377252959, −8.719215836141098887397849889482, −6.43862822492288780644749231806, −5.44427545868262519784083937980, −4.05034465450757557646608380672, −0.971485466404242146582954244011,
0.971485466404242146582954244011, 4.05034465450757557646608380672, 5.44427545868262519784083937980, 6.43862822492288780644749231806, 8.719215836141098887397849889482, 10.11740635326102555321377252959, 11.60289090716383510105410473431, 12.39443188719093491279321934128, 13.83719169395485278432700734985, 14.72417160241386605255424390681
