| L(s) = 1 | + (−4.89 + 2.82i)2-s + (−32.0 + 18.5i)3-s + (15.9 − 27.7i)4-s + (−25.9 − 44.9i)5-s + (104. − 181. i)6-s − 644.·7-s + 181. i·8-s + (321. − 557. i)9-s + (254. + 146. i)10-s + 1.88e3·11-s + 1.18e3i·12-s + (2.24e3 + 1.29e3i)13-s + (3.15e3 − 1.82e3i)14-s + (1.66e3 + 960. i)15-s + (−512. − 886. i)16-s + (−3.46e3 − 5.99e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−1.18 + 0.686i)3-s + (0.249 − 0.433i)4-s + (−0.207 − 0.359i)5-s + (0.485 − 0.840i)6-s − 1.87·7-s + 0.353i·8-s + (0.441 − 0.764i)9-s + (0.254 + 0.146i)10-s + 1.41·11-s + 0.686i·12-s + (1.02 + 0.589i)13-s + (1.15 − 0.664i)14-s + (0.493 + 0.284i)15-s + (−0.125 − 0.216i)16-s + (−0.704 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.518842 - 0.0275518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518842 - 0.0275518i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 19 | \( 1 + (-3.38e3 - 5.96e3i)T \) |
| good | 3 | \( 1 + (32.0 - 18.5i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (25.9 + 44.9i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 644.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.88e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-2.24e3 - 1.29e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (3.46e3 + 5.99e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (4.94e3 - 8.56e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.22e4 - 7.07e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 2.63e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 8.68e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (2.20e3 - 1.27e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (5.38e3 + 9.32e3i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.00e4 - 3.47e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-4.09e4 - 2.36e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (7.11e4 - 4.10e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-8.59e4 + 1.48e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.50e5 - 1.44e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.89e5 + 1.09e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-2.54e4 - 4.41e4i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (2.41e5 - 1.39e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 5.45e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (2.72e5 + 1.57e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-5.13e5 + 2.96e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−15.89017582255167783857993161521, −13.89080067255609384850787020024, −12.24425770792401486327005509916, −11.25295103354508224191845608050, −9.835141800943746083908256475003, −9.098423228466954331217853674421, −6.75837704707073924610783413510, −5.91214274565138405316277694390, −3.96501476924416937649846051203, −0.51927127556220499212215393926,
0.924552797410353258145755070293, 3.44411240333076977714774921493, 6.34821789565492876366201060477, 6.71278475565585279839885069217, 8.864482259984417150183718711047, 10.29516143755312897743339608264, 11.40262365810656972539135042444, 12.45733720755660892424171383472, 13.31108164034327039138921582708, 15.46507999390031026784235920514
