| L(s) = 1 | + (2.07 + 1.50i)2-s + (0.186 − 0.574i)3-s + (−0.439 − 1.35i)4-s + (11.4 − 8.30i)5-s + (1.25 − 0.910i)6-s + (−2.16 − 6.65i)7-s + (7.46 − 22.9i)8-s + (21.5 + 15.6i)9-s + 36.2·10-s + (−17.4 + 32.0i)11-s − 0.858·12-s + (−0.0838 − 0.0608i)13-s + (5.54 − 17.0i)14-s + (−2.63 − 8.11i)15-s + (40.9 − 29.7i)16-s + (23.8 − 17.3i)17-s + ⋯ |
| L(s) = 1 | + (0.733 + 0.533i)2-s + (0.0359 − 0.110i)3-s + (−0.0549 − 0.168i)4-s + (1.02 − 0.742i)5-s + (0.0853 − 0.0619i)6-s + (−0.116 − 0.359i)7-s + (0.330 − 1.01i)8-s + (0.798 + 0.579i)9-s + 1.14·10-s + (−0.477 + 0.878i)11-s − 0.0206·12-s + (−0.00178 − 0.00129i)13-s + (0.105 − 0.325i)14-s + (−0.0454 − 0.139i)15-s + (0.639 − 0.464i)16-s + (0.339 − 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.38121 - 0.191276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38121 - 0.191276i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (17.4 - 32.0i)T \) |
| good | 2 | \( 1 + (-2.07 - 1.50i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-0.186 + 0.574i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-11.4 + 8.30i)T + (38.6 - 118. i)T^{2} \) |
| 13 | \( 1 + (0.0838 + 0.0608i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-23.8 + 17.3i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (24.1 - 74.4i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (45.8 + 141. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-177. - 128. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-5.49 - 16.9i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (123. - 379. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (4.05 - 12.4i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-94.4 - 68.6i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-217. - 668. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-436. + 317. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-557. + 404. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-31.2 - 96.3i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (645. + 468. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-737. + 535. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 249.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-146. - 106. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.72235766721818071299327591626, −13.24971873141103800961427172241, −12.30154278313922384893919351233, −10.08059957605616145345814598601, −9.897359720362914424883834924694, −7.938385720432936647864544387701, −6.58340016375903398042424205854, −5.35698438335425578955390028534, −4.37391027357741957430918754129, −1.62852362734688257599069469727,
2.33452709157875041786077315945, 3.65866558170204135826772065350, 5.37346403539126421469324168963, 6.64533457322822910628098518740, 8.359806043872366293858589495521, 9.788002940272226997149506797000, 10.78369936182911600264659933482, 11.97688685368451576732997906416, 13.07206027168631081013947462716, 13.76954652786640509047216674422
