| L(s) = 1 | + 3.46·2-s + 3.98·4-s − 6.55·5-s − 13.9·8-s − 22.6·10-s − 2.29·11-s − 28.6·13-s − 79.9·16-s + 46.6·17-s + 67.5·19-s − 26.0·20-s − 7.94·22-s + 30.1·23-s − 82.0·25-s − 99.1·26-s − 24.0·29-s + 193.·31-s − 165.·32-s + 161.·34-s + 208.·37-s + 233.·38-s + 91.2·40-s + 234.·41-s + 46.0·43-s − 9.13·44-s + 104.·46-s − 194.·47-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.497·4-s − 0.586·5-s − 0.614·8-s − 0.717·10-s − 0.0628·11-s − 0.610·13-s − 1.24·16-s + 0.665·17-s + 0.815·19-s − 0.291·20-s − 0.0769·22-s + 0.273·23-s − 0.656·25-s − 0.747·26-s − 0.154·29-s + 1.12·31-s − 0.914·32-s + 0.814·34-s + 0.927·37-s + 0.997·38-s + 0.360·40-s + 0.892·41-s + 0.163·43-s − 0.0312·44-s + 0.334·46-s − 0.602·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.966383225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.966383225\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 5 | \( 1 + 6.55T + 125T^{2} \) |
| 11 | \( 1 + 2.29T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 208.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 221.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 710.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 634.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 234.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 213.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 242.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−9.392500739924377522686536459460, −8.262703698089704853361811518407, −7.55167804982042962625579931414, −6.61540516872600281911218896094, −5.68565205440530252982700229046, −4.97177094836350747381239271623, −4.13521661008748257302693917788, −3.33899525515158607208811248250, −2.42838695215584670111836134926, −0.70745995895241811912258324403,
0.70745995895241811912258324403, 2.42838695215584670111836134926, 3.33899525515158607208811248250, 4.13521661008748257302693917788, 4.97177094836350747381239271623, 5.68565205440530252982700229046, 6.61540516872600281911218896094, 7.55167804982042962625579931414, 8.262703698089704853361811518407, 9.392500739924377522686536459460
