| L(s) = 1 | + 1.51i·3-s + 15.8·5-s + 74.7·7-s + 78.7·9-s − 8.49·11-s − 283. i·13-s + 23.9i·15-s − 36.0·17-s + (345. + 104. i)19-s + 113. i·21-s − 488.·23-s − 375.·25-s + 241. i·27-s − 492. i·29-s + 1.28e3i·31-s + ⋯ |
| L(s) = 1 | + 0.168i·3-s + 0.632·5-s + 1.52·7-s + 0.971·9-s − 0.0702·11-s − 1.68i·13-s + 0.106i·15-s − 0.124·17-s + (0.957 + 0.288i)19-s + 0.256i·21-s − 0.923·23-s − 0.600·25-s + 0.331i·27-s − 0.585i·29-s + 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.903408471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903408471\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-345. - 104. i)T \) |
| good | 3 | \( 1 - 1.51iT - 81T^{2} \) |
| 5 | \( 1 - 15.8T + 625T^{2} \) |
| 7 | \( 1 - 74.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 8.49T + 1.46e4T^{2} \) |
| 13 | \( 1 + 283. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 36.0T + 8.35e4T^{2} \) |
| 23 | \( 1 + 488.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 492. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.28e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.13e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.07e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.05e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.21e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 4.68e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.20e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.16e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.86e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.26e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.39e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.31e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 672.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.11e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.06e4iT - 8.85e7T^{2} \) |
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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
−10.78139902151766616369627695056, −10.28162012845148012604881824011, −9.230323297900177016668582358247, −7.983601472362968270478220801276, −7.45329193203621185856783494964, −5.80175883057918772567679889741, −5.10175060957036494686038407083, −3.87895261526777080941433914016, −2.20498538306078922519737096524, −1.04798330473150124810345572423,
1.37056151709655317818985439741, 2.10531269556153519018220307282, 4.15217741499981980194652200861, 4.94698179282126717541254477886, 6.23801877505111240643404490102, 7.31001409473013800505353136625, 8.166952795095867717580007980065, 9.366347385325897546329356479515, 10.05578504664505416804676049753, 11.40052818054186771771602422376
