| L(s) = 1 | − 19.6·2-s − 27·3-s + 257.·4-s − 311.·5-s + 530.·6-s + 487.·7-s − 2.54e3·8-s + 729·9-s + 6.12e3·10-s − 3.55e3·11-s − 6.95e3·12-s − 5.49e3·13-s − 9.57e3·14-s + 8.41e3·15-s + 1.70e4·16-s − 3.68e4·17-s − 1.43e4·18-s − 2.28e4·19-s − 8.02e4·20-s − 1.31e4·21-s + 6.98e4·22-s − 1.21e4·23-s + 6.87e4·24-s + 1.90e4·25-s + 1.07e5·26-s − 1.96e4·27-s + 1.25e5·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.577·3-s + 2.01·4-s − 1.11·5-s + 1.00·6-s + 0.537·7-s − 1.75·8-s + 0.333·9-s + 1.93·10-s − 0.805·11-s − 1.16·12-s − 0.693·13-s − 0.932·14-s + 0.643·15-s + 1.03·16-s − 1.82·17-s − 0.578·18-s − 0.763·19-s − 2.24·20-s − 0.310·21-s + 1.39·22-s − 0.208·23-s + 1.01·24-s + 0.243·25-s + 1.20·26-s − 0.192·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2108822682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2108822682\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 19.6T + 128T^{2} \) |
| 5 | \( 1 + 311.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 487.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.49e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.68e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.28e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 4.98e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.53e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.02e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.81e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.07e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.17e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.96e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.04e7T + 8.07e13T^{2} \) |
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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
−12.87402716138589204918789729062, −11.46549662580964954587768269635, −11.07957371262368366239296426086, −9.885077844599755428161751538190, −8.490285516291160737803526991772, −7.72788290660796947904543955836, −6.63078020618949735207287251312, −4.57881333564037658735141246070, −2.20629013659206182602283683395, −0.39170983389836999141453437240,
0.39170983389836999141453437240, 2.20629013659206182602283683395, 4.57881333564037658735141246070, 6.63078020618949735207287251312, 7.72788290660796947904543955836, 8.490285516291160737803526991772, 9.885077844599755428161751538190, 11.07957371262368366239296426086, 11.46549662580964954587768269635, 12.87402716138589204918789729062
