| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−9.32 + 10.3i)3-s + (−12.9 − 9.40i)4-s + (3.76 − 6.51i)5-s + (−27.8 − 48.2i)6-s + (24.2 − 230. i)7-s + (51.7 − 37.6i)8-s + (5.12 + 48.7i)9-s + (20.1 + 22.3i)10-s + (−104. + 46.4i)11-s + (217. − 46.3i)12-s + (669. + 142. i)13-s + (848. + 377. i)14-s + (32.3 + 99.6i)15-s + (79.1 + 243. i)16-s + (−1.07e3 − 478. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.597 + 0.664i)3-s + (−0.404 − 0.293i)4-s + (0.0672 − 0.116i)5-s + (−0.315 − 0.547i)6-s + (0.187 − 1.78i)7-s + (0.286 − 0.207i)8-s + (0.0210 + 0.200i)9-s + (0.0636 + 0.0707i)10-s + (−0.259 + 0.115i)11-s + (0.437 − 0.0928i)12-s + (1.09 + 0.233i)13-s + (1.15 + 0.514i)14-s + (0.0371 + 0.114i)15-s + (0.0772 + 0.237i)16-s + (−0.901 − 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.16100 + 0.0653452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16100 + 0.0653452i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 31 | \( 1 + (-4.82e3 + 2.31e3i)T \) |
| good | 3 | \( 1 + (9.32 - 10.3i)T + (-25.4 - 241. i)T^{2} \) |
| 5 | \( 1 + (-3.76 + 6.51i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-24.2 + 230. i)T + (-1.64e4 - 3.49e3i)T^{2} \) |
| 11 | \( 1 + (104. - 46.4i)T + (1.07e5 - 1.19e5i)T^{2} \) |
| 13 | \( 1 + (-669. - 142. i)T + (3.39e5 + 1.51e5i)T^{2} \) |
| 17 | \( 1 + (1.07e3 + 478. i)T + (9.50e5 + 1.05e6i)T^{2} \) |
| 19 | \( 1 + (-2.55e3 + 542. i)T + (2.26e6 - 1.00e6i)T^{2} \) |
| 23 | \( 1 + (-1.07e3 + 784. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.94e3 + 5.99e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 37 | \( 1 + (1.42e3 + 2.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (9.53e3 + 1.05e4i)T + (-1.21e7 + 1.15e8i)T^{2} \) |
| 43 | \( 1 + (-1.20e4 + 2.55e3i)T + (1.34e8 - 5.97e7i)T^{2} \) |
| 47 | \( 1 + (-2.77e3 - 8.55e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.77e3 + 1.68e4i)T + (-4.09e8 + 8.69e7i)T^{2} \) |
| 59 | \( 1 + (-9.74e3 + 1.08e4i)T + (-7.47e7 - 7.11e8i)T^{2} \) |
| 61 | \( 1 + 1.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.38e3 + 4.13e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.43e3 - 4.21e4i)T + (-1.76e9 + 3.75e8i)T^{2} \) |
| 73 | \( 1 + (-5.49e4 + 2.44e4i)T + (1.38e9 - 1.54e9i)T^{2} \) |
| 79 | \( 1 + (-8.24e4 - 3.67e4i)T + (2.05e9 + 2.28e9i)T^{2} \) |
| 83 | \( 1 + (3.41e4 + 3.78e4i)T + (-4.11e8 + 3.91e9i)T^{2} \) |
| 89 | \( 1 + (-5.39e4 - 3.91e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (5.80e4 + 4.22e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.80702890903546639110395980452, −13.48788736039692300383445978051, −11.31259241953981818854838219490, −10.58559176640677342049288445059, −9.489505380128808875345282361314, −7.87413505037003357923725301649, −6.76771355334577090265923566712, −5.14039121224454330013344687423, −4.06840802770253005745875958110, −0.75946550164543193407654049260,
1.30899531351196521738536386569, 2.99139240418713537492925692817, 5.32087205978742329126496679212, 6.46557300461811608925281340264, 8.307241417479570030755593299388, 9.257940403325825218136651019066, 10.86744416740761871105371325912, 11.87607270756708626374423666903, 12.47381721462878237046917890170, 13.63979910420120269449854785834
