| L(s) = 1 | + (−74.2 − 32.2i)3-s + 4.01e3·7-s + (4.47e3 + 4.79e3i)9-s + 1.13e4i·11-s − 1.46e4·13-s + 1.21e5i·17-s − 2.28e4·19-s + (−2.98e5 − 1.29e5i)21-s + 4.52e5i·23-s + (−1.77e5 − 5.00e5i)27-s + 9.25e5i·29-s − 1.24e6·31-s + (3.65e5 − 8.41e5i)33-s − 9.96e3·37-s + (1.08e6 + 4.73e5i)39-s + ⋯ |
| L(s) = 1 | + (−0.917 − 0.398i)3-s + 1.67·7-s + (0.682 + 0.730i)9-s + 0.774i·11-s − 0.513·13-s + 1.45i·17-s − 0.175·19-s + (−1.53 − 0.666i)21-s + 1.61i·23-s + (−0.334 − 0.942i)27-s + 1.30i·29-s − 1.34·31-s + (0.308 − 0.709i)33-s − 0.00531·37-s + (0.471 + 0.204i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7032464286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7032464286\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (74.2 + 32.2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.01e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.13e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.46e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.21e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.28e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.52e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 9.25e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.24e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 9.96e3T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.24e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 9.86e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 7.49e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.98e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.17e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.04e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.35e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.35e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.39e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.08e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.54e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.17e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.22e8T + 7.83e15T^{2} \) |
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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
−10.90859520961323950020627766209, −10.10369561958976257909911195657, −8.714000582302237748793407503797, −7.67170667540411824002236784518, −7.07210461853707951216798362208, −5.63349019755431532803543956566, −5.02164829221173362576780237689, −3.96590809011279451850908548291, −1.86457692166942103550724537827, −1.49703225929375061118708820627,
0.17008238242979917882737652424, 1.15207599860035991413251206458, 2.54483992584877400754739461886, 4.22790758925207162080854430519, 4.92339765550064986341026803902, 5.77442106638232119489176665818, 7.00033108081839369442066590540, 8.005023053672468602917821149318, 9.023788940175374408475831824874, 10.13330257111018629054254392071
