| L(s) = 1 | + (3.73 + 1.15i)2-s + (0.0993 − 0.253i)3-s + (−13.7 − 9.40i)4-s + (41.9 − 6.31i)5-s + (0.663 − 0.831i)6-s + (64.1 − 112. i)7-s + (−118. − 148. i)8-s + (178. + 165. i)9-s + (163. + 24.7i)10-s + (391. − 363. i)11-s + (−3.74 + 2.55i)12-s + (−122. − 538. i)13-s + (369. − 347. i)14-s + (2.56 − 11.2i)15-s + (−77.2 − 196. i)16-s + (95.5 + 1.27e3i)17-s + ⋯ |
| L(s) = 1 | + (0.661 + 0.203i)2-s + (0.00637 − 0.0162i)3-s + (−0.430 − 0.293i)4-s + (0.749 − 0.112i)5-s + (0.00752 − 0.00943i)6-s + (0.494 − 0.868i)7-s + (−0.656 − 0.822i)8-s + (0.732 + 0.679i)9-s + (0.518 + 0.0781i)10-s + (0.976 − 0.905i)11-s + (−0.00751 + 0.00512i)12-s + (−0.201 − 0.883i)13-s + (0.504 − 0.473i)14-s + (0.00294 − 0.0128i)15-s + (−0.0754 − 0.192i)16-s + (0.0802 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.639i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.25450 - 0.814444i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25450 - 0.814444i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-64.1 + 112. i)T \) |
| good | 2 | \( 1 + (-3.73 - 1.15i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (-0.0993 + 0.253i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (-41.9 + 6.31i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (-391. + 363. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (122. + 538. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-95.5 - 1.27e3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (8.21 - 14.2i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-113. + 1.50e3i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (-429. - 206. i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-626. - 1.08e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.06e4 - 7.27e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (-299. - 376. i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (1.08e4 - 1.36e4i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-2.69e4 - 8.32e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (-3.57e3 - 2.43e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (4.15e3 + 626. i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-3.24e4 + 2.21e4i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-3.06e3 - 5.31e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.57e4 + 1.72e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-7.97e4 + 2.46e4i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (2.44e4 - 4.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.06e4 - 9.06e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (3.78e4 + 3.51e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + 6.12e4T + 8.58e9T^{2} \) |
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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
−14.15711319221941588049961319326, −13.59149247679440309556062125073, −12.59590118109603670608337804241, −10.75486250154327745060901832952, −9.833835926499661077909563263861, −8.284719127326225735850323260778, −6.53434833087724297999880256011, −5.23342146430829724144135897211, −3.89749565791239143722011769702, −1.21253006039701835684817270298,
2.05944719887419180759752343761, 4.04998484959174906386901974179, 5.38323920735162538804524934603, 6.97714919021540358589903622239, 8.970519761192199249017136899225, 9.662658048151002407375645423006, 11.78827221310031982531281689301, 12.27851586778622434492740902177, 13.68382809072439294007876181912, 14.47145599793284236466750592549
