| 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
−14.47145599793284236466750592549, −13.68382809072439294007876181912, −12.27851586778622434492740902177, −11.78827221310031982531281689301, −9.662658048151002407375645423006, −8.970519761192199249017136899225, −6.97714919021540358589903622239, −5.38323920735162538804524934603, −4.04998484959174906386901974179, −2.05944719887419180759752343761,
1.21253006039701835684817270298, 3.89749565791239143722011769702, 5.23342146430829724144135897211, 6.53434833087724297999880256011, 8.284719127326225735850323260778, 9.833835926499661077909563263861, 10.75486250154327745060901832952, 12.59590118109603670608337804241, 13.59149247679440309556062125073, 14.15711319221941588049961319326
