| L(s) = 1 | + (2.82 − 2.82i)2-s − 16.0i·4-s + (−140. − 140. i)7-s + (−45.2 − 45.2i)8-s + 540. i·11-s + (798. − 798. i)13-s − 795.·14-s − 256.·16-s + (598. − 598. i)17-s − 1.78e3i·19-s + (1.53e3 + 1.53e3i)22-s + (−718. − 718. i)23-s − 4.51e3i·26-s + (−2.25e3 + 2.25e3i)28-s − 5.38e3·29-s + ⋯ |
| L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−1.08 − 1.08i)7-s + (−0.250 − 0.250i)8-s + 1.34i·11-s + (1.31 − 1.31i)13-s − 1.08·14-s − 0.250·16-s + (0.502 − 0.502i)17-s − 1.13i·19-s + (0.673 + 0.673i)22-s + (−0.283 − 0.283i)23-s − 1.31i·26-s + (−0.542 + 0.542i)28-s − 1.18·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8420778564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8420778564\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (140. + 140. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 540. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-798. + 798. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-598. + 598. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.78e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (718. + 718. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 5.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (7.34e3 + 7.34e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.24e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (9.90e3 - 9.90e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.82e3 - 1.82e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.60e4 - 1.60e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 4.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 605.T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.30e4 - 2.30e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.13e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (4.18e4 - 4.18e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 6.19e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (8.21e4 + 8.21e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.14e4 + 5.14e4i)T + 8.58e9iT^{2} \) |
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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
−10.00377922191097231729672085314, −9.120887942037106250703929343569, −7.68941128340057254046239352707, −6.87991644162360042130193646874, −5.89620127784014623797805843974, −4.69874685837482107559798656806, −3.69617581536596817922185714501, −2.85428733993800861806030775113, −1.26930785297220474973996332047, −0.16849799882825601097644230021,
1.73360306205929097678624102107, 3.29634356520937192924723349119, 3.82458537804834153518726370274, 5.60664154367927179838656325853, 5.99943801716267643783016054351, 6.84064309950924810925167425563, 8.327111718676254490507949536759, 8.785182509470208564110018062628, 9.782293959099783041203312704674, 11.01946554025863391696481436277
