| L(s) = 1 | + 2·2-s + 4.85·3-s + 4·4-s + 13.8·5-s + 9.71·6-s − 7·7-s + 8·8-s − 3.39·9-s + 27.6·10-s + 51.8·11-s + 19.4·12-s − 14·14-s + 67.2·15-s + 16·16-s − 105.·17-s − 6.79·18-s − 48.0·19-s + 55.3·20-s − 34.0·21-s + 103.·22-s + 22.8·23-s + 38.8·24-s + 66.6·25-s − 147.·27-s − 28·28-s + 233.·29-s + 134.·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.935·3-s + 0.5·4-s + 1.23·5-s + 0.661·6-s − 0.377·7-s + 0.353·8-s − 0.125·9-s + 0.875·10-s + 1.42·11-s + 0.467·12-s − 0.267·14-s + 1.15·15-s + 0.250·16-s − 1.50·17-s − 0.0889·18-s − 0.579·19-s + 0.619·20-s − 0.353·21-s + 1.00·22-s + 0.207·23-s + 0.330·24-s + 0.532·25-s − 1.05·27-s − 0.188·28-s + 1.49·29-s + 0.818·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.913570397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.913570397\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4.85T + 27T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 11 | \( 1 - 51.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 72.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 320.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 343.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 147.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 779.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 962.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 355.T + 9.12e5T^{2} \) |
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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
−8.928817008330313148804380064487, −7.913152434985545420071680723974, −6.63230747865330644917204880995, −6.46299305556435265890971565897, −5.58777055477804964956959988489, −4.43218996637030562210752056686, −3.81361603470575608144026749623, −2.55490594178410214649610257430, −2.31171489352338095683790263841, −1.03760229360448849598560500332,
1.03760229360448849598560500332, 2.31171489352338095683790263841, 2.55490594178410214649610257430, 3.81361603470575608144026749623, 4.43218996637030562210752056686, 5.58777055477804964956959988489, 6.46299305556435265890971565897, 6.63230747865330644917204880995, 7.913152434985545420071680723974, 8.928817008330313148804380064487
