| L(s) = 1 | + (−33.0 + 33.0i)3-s + (1.42e3 + 1.42e3i)7-s − 2.18e3i·9-s − 6.44e3·11-s + (1.40e4 − 1.40e4i)13-s + (−2.19e4 − 2.19e4i)17-s + 3.30e4i·19-s − 9.40e4·21-s + (−2.30e4 + 2.30e4i)23-s + (7.23e4 + 7.23e4i)27-s + 5.41e5i·29-s − 5.71e5·31-s + (2.13e5 − 2.13e5i)33-s + (9.45e5 + 9.45e5i)37-s + 9.27e5i·39-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.592 + 0.592i)7-s − 0.333i·9-s − 0.440·11-s + (0.491 − 0.491i)13-s + (−0.262 − 0.262i)17-s + 0.253i·19-s − 0.483·21-s + (−0.0824 + 0.0824i)23-s + (0.136 + 0.136i)27-s + 0.766i·29-s − 0.618·31-s + (0.179 − 0.179i)33-s + (0.504 + 0.504i)37-s + 0.401i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3570924402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3570924402\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (33.0 - 33.0i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-1.42e3 - 1.42e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + 6.44e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-1.40e4 + 1.40e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (2.19e4 + 2.19e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 3.30e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (2.30e4 - 2.30e4i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 - 5.41e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 5.71e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-9.45e5 - 9.45e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 4.61e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (-2.14e6 + 2.14e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-3.75e6 - 3.75e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (-1.84e5 + 1.84e5i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 - 6.00e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.56e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-9.12e6 - 9.12e6i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 - 1.50e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (3.53e7 - 3.53e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 5.63e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (3.49e7 - 3.49e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 8.31e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (6.21e7 + 6.21e7i)T + 7.83e15iT^{2} \) |
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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
−10.87125947047059621484548121755, −10.03559390863047044220617598568, −8.920961925322185285452371095955, −8.145570384402092741929367445363, −6.95310885552081892446642556866, −5.71398273051461255759253515813, −5.08813787159060326084869375584, −3.86545538343842305771819547490, −2.61704976206270438812431291765, −1.29896010282164829420887538834,
0.081092779207097675407643719895, 1.22276292459379056402265492204, 2.28232676516454904757261536148, 3.84044483774001763807442511261, 4.84985484022673189783881999341, 5.94816070251025669472806914210, 6.96420807602142869954838644987, 7.83239928686873645547572579161, 8.757483279608826656592445714704, 10.00066094129938698911725804056
