| L(s) = 1 | + 1.16·3-s + 16.7·5-s − 31.7·7-s − 25.6·9-s + 10.5·11-s + 53.0·13-s + 19.5·15-s + 17·17-s + 26.4·19-s − 37.0·21-s − 200.·23-s + 156.·25-s − 61.3·27-s + 4.00·29-s + 243.·31-s + 12.3·33-s − 532.·35-s + 158.·37-s + 61.8·39-s + 360.·41-s + 476.·43-s − 430.·45-s − 418.·47-s + 664.·49-s + 19.8·51-s + 110.·53-s + 177.·55-s + ⋯ |
| L(s) = 1 | + 0.224·3-s + 1.50·5-s − 1.71·7-s − 0.949·9-s + 0.289·11-s + 1.13·13-s + 0.336·15-s + 0.242·17-s + 0.319·19-s − 0.384·21-s − 1.82·23-s + 1.25·25-s − 0.437·27-s + 0.0256·29-s + 1.40·31-s + 0.0649·33-s − 2.57·35-s + 0.703·37-s + 0.253·39-s + 1.37·41-s + 1.69·43-s − 1.42·45-s − 1.29·47-s + 1.93·49-s + 0.0544·51-s + 0.285·53-s + 0.434·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.386756087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386756087\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 1.16T + 27T^{2} \) |
| 5 | \( 1 - 16.7T + 125T^{2} \) |
| 7 | \( 1 + 31.7T + 343T^{2} \) |
| 11 | \( 1 - 10.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.0T + 2.19e3T^{2} \) |
| 19 | \( 1 - 26.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.00T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 110.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 31.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 124.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 38.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 978.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 391.T + 9.12e5T^{2} \) |
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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
−9.613523441317652628046461185856, −8.898665373716693271517452598063, −7.981053164270230517101413223296, −6.57510076239484593705176424442, −6.06014432190774226163313564638, −5.68533731155379999522007943054, −4.01427190769473430056802920412, −3.04019662764045414392944009040, −2.24711681279619361585865991320, −0.789802972027727871278799511564,
0.789802972027727871278799511564, 2.24711681279619361585865991320, 3.04019662764045414392944009040, 4.01427190769473430056802920412, 5.68533731155379999522007943054, 6.06014432190774226163313564638, 6.57510076239484593705176424442, 7.981053164270230517101413223296, 8.898665373716693271517452598063, 9.613523441317652628046461185856
