| L(s) = 1 | − 164.·5-s + 343·7-s − 4.56e3·11-s − 2.03e3·13-s + 1.73e4·17-s + 5.88e3·19-s − 4.11e4·23-s − 5.11e4·25-s − 1.41e5·29-s + 1.70e5·31-s − 5.63e4·35-s + 2.25e5·37-s − 6.94e5·41-s − 7.49e5·43-s + 1.34e6·47-s + 1.17e5·49-s + 1.79e6·53-s + 7.50e5·55-s − 1.15e5·59-s − 9.79e5·61-s + 3.33e5·65-s − 4.25e6·67-s − 4.24e6·71-s + 2.29e5·73-s − 1.56e6·77-s − 3.19e6·79-s + 8.78e6·83-s + ⋯ |
| L(s) = 1 | − 0.588·5-s + 0.377·7-s − 1.03·11-s − 0.256·13-s + 0.856·17-s + 0.196·19-s − 0.705·23-s − 0.654·25-s − 1.07·29-s + 1.02·31-s − 0.222·35-s + 0.730·37-s − 1.57·41-s − 1.43·43-s + 1.89·47-s + 0.142·49-s + 1.65·53-s + 0.608·55-s − 0.0731·59-s − 0.552·61-s + 0.150·65-s − 1.72·67-s − 1.40·71-s + 0.0691·73-s − 0.391·77-s − 0.728·79-s + 1.68·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.366487034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366487034\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 + 164.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.03e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.88e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.11e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.94e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.34e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.79e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.15e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.79e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.25e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.29e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.74e6T + 8.07e13T^{2} \) |
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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
−9.968022834331195005435139787813, −8.738524595264386810773886437790, −7.84191976500565444556909852103, −7.37924430293914952244732520301, −5.97006116866405071617658885871, −5.11675979908806819160122867837, −4.08105576817768873951577048490, −3.01965342867615853851841285316, −1.85690778642305017952533147172, −0.50217612421704336489027708455,
0.50217612421704336489027708455, 1.85690778642305017952533147172, 3.01965342867615853851841285316, 4.08105576817768873951577048490, 5.11675979908806819160122867837, 5.97006116866405071617658885871, 7.37924430293914952244732520301, 7.84191976500565444556909852103, 8.738524595264386810773886437790, 9.968022834331195005435139787813
