| L(s) = 1 | − 2.38·2-s + 27·3-s − 122.·4-s − 147.·5-s − 64.5·6-s − 1.22e3·7-s + 598.·8-s + 729·9-s + 351.·10-s − 1.63e3·11-s − 3.30e3·12-s + 1.28e4·13-s + 2.92e3·14-s − 3.97e3·15-s + 1.42e4·16-s − 1.20e4·17-s − 1.74e3·18-s + 2.76e4·19-s + 1.80e4·20-s − 3.30e4·21-s + 3.91e3·22-s + 1.21e4·23-s + 1.61e4·24-s − 5.64e4·25-s − 3.08e4·26-s + 1.96e4·27-s + 1.49e5·28-s + ⋯ |
| L(s) = 1 | − 0.211·2-s + 0.577·3-s − 0.955·4-s − 0.526·5-s − 0.121·6-s − 1.34·7-s + 0.412·8-s + 0.333·9-s + 0.111·10-s − 0.370·11-s − 0.551·12-s + 1.62·13-s + 0.284·14-s − 0.304·15-s + 0.868·16-s − 0.594·17-s − 0.0703·18-s + 0.923·19-s + 0.503·20-s − 0.778·21-s + 0.0783·22-s + 0.208·23-s + 0.238·24-s − 0.722·25-s − 0.343·26-s + 0.192·27-s + 1.28·28-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
| \(L(4)\) |
\(\approx\) |
\(1.206813035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206813035\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 2.38T + 128T^{2} \) |
| 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
−13.36041411690481770344856388670, −12.51449469964901707508529569178, −10.82431439453945624008667708173, −9.592967434654108529882477059570, −8.789766875480945065665705634362, −7.65368108407479921717213724611, −6.06360193463856057177354323346, −4.19928359499263201926534822281, −3.16155955781121281193469615631, −0.75756086151303984421225031355,
0.75756086151303984421225031355, 3.16155955781121281193469615631, 4.19928359499263201926534822281, 6.06360193463856057177354323346, 7.65368108407479921717213724611, 8.789766875480945065665705634362, 9.592967434654108529882477059570, 10.82431439453945624008667708173, 12.51449469964901707508529569178, 13.36041411690481770344856388670
