| L(s) = 1 | + (2.46 + 1.78i)2-s + (1.79 − 5.51i)3-s + (0.388 + 1.19i)4-s + (2.07 − 1.50i)5-s + (14.2 − 10.3i)6-s + (2.16 + 6.65i)7-s + (6.33 − 19.5i)8-s + (−5.40 − 3.92i)9-s + 7.78·10-s + (25.8 − 25.7i)11-s + 7.30·12-s + (−0.799 − 0.580i)13-s + (−6.58 + 20.2i)14-s + (−4.58 − 14.1i)15-s + (58.6 − 42.6i)16-s + (−67.4 + 49.0i)17-s + ⋯ |
| L(s) = 1 | + (0.870 + 0.632i)2-s + (0.345 − 1.06i)3-s + (0.0485 + 0.149i)4-s + (0.185 − 0.134i)5-s + (0.972 − 0.706i)6-s + (0.116 + 0.359i)7-s + (0.280 − 0.862i)8-s + (−0.200 − 0.145i)9-s + 0.246·10-s + (0.708 − 0.705i)11-s + 0.175·12-s + (−0.0170 − 0.0123i)13-s + (−0.125 + 0.386i)14-s + (−0.0789 − 0.243i)15-s + (0.916 − 0.665i)16-s + (−0.962 + 0.699i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.49675 - 0.571040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49675 - 0.571040i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.16 - 6.65i)T \) |
| 11 | \( 1 + (-25.8 + 25.7i)T \) |
| good | 2 | \( 1 + (-2.46 - 1.78i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.79 + 5.51i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-2.07 + 1.50i)T + (38.6 - 118. i)T^{2} \) |
| 13 | \( 1 + (0.799 + 0.580i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (67.4 - 49.0i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (49.8 - 153. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + 16.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-51.6 - 159. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (80.0 + 58.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (88.8 + 273. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-50.0 + 154. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 333.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (112. - 346. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-37.8 - 27.5i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (47.3 + 145. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-84.6 + 61.5i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 13.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-403. + 293. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (264. + 815. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-450. - 327. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (571. - 414. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 268.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.21e3 + 883. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−14.03023385165927355734057407646, −12.94660899014399193411245871313, −12.37053912572831865407838578694, −10.71358225683619296709528595137, −9.078048055259206260595397351583, −7.81937042036803047206397836997, −6.56481045092894021594630931996, −5.70200146323922405306641152972, −3.93298476432657172231181956053, −1.63080503642366251251406703297,
2.57037106511855445948883571703, 4.13707818888711723178198449569, 4.75464870103350016247236785268, 6.84159457128732593519626645111, 8.659664207744815745426264140613, 9.753578341760256989409531223272, 10.88763193593706650911952478026, 11.83321578130161109887529111990, 13.11118632963059752242634710956, 13.98204462571990356876541111649
