L(s) = 1 | + 2.38·2-s − 122.·4-s + 147.·5-s − 1.22e3·7-s − 598.·8-s + 351.·10-s + 1.63e3·11-s + 1.28e4·13-s − 2.92e3·14-s + 1.42e4·16-s + 1.20e4·17-s + 2.76e4·19-s − 1.80e4·20-s + 3.91e3·22-s − 1.21e4·23-s − 5.64e4·25-s + 3.08e4·26-s + 1.49e5·28-s − 2.40e5·29-s − 1.28e5·31-s + 1.10e5·32-s + 2.87e4·34-s − 1.80e5·35-s + 1.79e5·37-s + 6.59e4·38-s − 8.80e4·40-s − 4.73e5·41-s + ⋯ |
L(s) = 1 | + 0.211·2-s − 0.955·4-s + 0.526·5-s − 1.34·7-s − 0.412·8-s + 0.111·10-s + 0.370·11-s + 1.62·13-s − 0.284·14-s + 0.868·16-s + 0.594·17-s + 0.923·19-s − 0.503·20-s + 0.0783·22-s − 0.208·23-s − 0.722·25-s + 0.343·26-s + 1.28·28-s − 1.83·29-s − 0.772·31-s + 0.596·32-s + 0.125·34-s − 0.710·35-s + 0.581·37-s + 0.195·38-s − 0.217·40-s − 1.07·41-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
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 - 2.38T + 128T^{2} \) |
| 5 | \( 1 - 147.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.22e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.28e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.76e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.40e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.73e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.96e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.90e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.41e6T + 8.07e13T^{2} \) |
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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
−10.41877314504141105099171507814, −9.450616585209084199615233642935, −9.022586273661011526304623965837, −7.60934497956253770263251169678, −6.12340142697998932575251433271, −5.63245545424066020410247212006, −3.93685072169896290415561195984, −3.26366021107754661680001025458, −1.33850475867955891696736088387, 0,
1.33850475867955891696736088387, 3.26366021107754661680001025458, 3.93685072169896290415561195984, 5.63245545424066020410247212006, 6.12340142697998932575251433271, 7.60934497956253770263251169678, 9.022586273661011526304623965837, 9.450616585209084199615233642935, 10.41877314504141105099171507814