| L(s) = 1 | − 19.4·2-s + 249.·4-s + 31.9·5-s − 461.·7-s − 2.35e3·8-s − 620.·10-s − 4.54e3·11-s − 1.11e4·13-s + 8.95e3·14-s + 1.38e4·16-s − 1.45e4·17-s + 2.07e4·19-s + 7.96e3·20-s + 8.81e4·22-s + 1.21e4·23-s − 7.71e4·25-s + 2.17e5·26-s − 1.14e5·28-s + 3.16e4·29-s + 4.55e4·31-s + 3.24e4·32-s + 2.82e5·34-s − 1.47e4·35-s − 3.79e4·37-s − 4.02e5·38-s − 7.52e4·40-s − 6.62e5·41-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.94·4-s + 0.114·5-s − 0.508·7-s − 1.62·8-s − 0.196·10-s − 1.02·11-s − 1.41·13-s + 0.872·14-s + 0.845·16-s − 0.717·17-s + 0.693·19-s + 0.222·20-s + 1.76·22-s + 0.208·23-s − 0.986·25-s + 2.42·26-s − 0.989·28-s + 0.241·29-s + 0.274·31-s + 0.175·32-s + 1.23·34-s − 0.0580·35-s − 0.123·37-s − 1.19·38-s − 0.185·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3076896522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3076896522\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 + 19.4T + 128T^{2} \) |
| 5 | \( 1 - 31.9T + 7.81e4T^{2} \) |
| 7 | \( 1 + 461.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.11e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.07e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 3.16e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.55e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.79e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.52e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.39e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.45e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.71e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.85e6T + 8.07e13T^{2} \) |
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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
−10.74736239549519417189565721230, −9.875475273409534431665710255564, −9.373777689291564793234507061987, −8.134179105944221221725341909634, −7.42802147560099517441808650679, −6.43539887964696163702942958138, −4.96856164216448259423334808806, −2.92547406380803039929463648047, −1.92527784374270019041383788551, −0.36023635747918167199098604055,
0.36023635747918167199098604055, 1.92527784374270019041383788551, 2.92547406380803039929463648047, 4.96856164216448259423334808806, 6.43539887964696163702942958138, 7.42802147560099517441808650679, 8.134179105944221221725341909634, 9.373777689291564793234507061987, 9.875475273409534431665710255564, 10.74736239549519417189565721230
