| L(s) = 1 | + (2.64 + 11.9i)2-s + (7.84 − 47.8i)3-s + (−78.8 + 36.4i)4-s + (37.8 − 136. i)5-s + (595. − 32.2i)6-s + (−396. + 375. i)7-s + (−170. − 223. i)8-s + (−1.53e3 − 518. i)9-s + (1.73e3 + 94.0i)10-s + (−129. + 109. i)11-s + (1.12e3 + 4.06e3i)12-s + (−952. − 2.82e3i)13-s + (−5.55e3 − 3.76e3i)14-s + (−6.22e3 − 2.88e3i)15-s + (−1.36e3 + 1.60e3i)16-s + (−6.30e3 − 5.97e3i)17-s + ⋯ |
| L(s) = 1 | + (0.330 + 1.49i)2-s + (0.290 − 1.77i)3-s + (−1.23 + 0.569i)4-s + (0.302 − 1.09i)5-s + (2.75 − 0.149i)6-s + (−1.15 + 1.09i)7-s + (−0.332 − 0.436i)8-s + (−2.11 − 0.711i)9-s + (1.73 + 0.0940i)10-s + (−0.0971 + 0.0825i)11-s + (0.652 + 2.35i)12-s + (−0.433 − 1.28i)13-s + (−2.02 − 1.37i)14-s + (−1.84 − 0.853i)15-s + (−0.333 + 0.392i)16-s + (−1.28 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.581494 - 0.819162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581494 - 0.819162i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-1.41e5 - 1.49e5i)T \) |
| good | 2 | \( 1 + (-2.64 - 11.9i)T + (-58.0 + 26.8i)T^{2} \) |
| 3 | \( 1 + (-7.84 + 47.8i)T + (-690. - 232. i)T^{2} \) |
| 5 | \( 1 + (-37.8 + 136. i)T + (-1.33e4 - 8.05e3i)T^{2} \) |
| 7 | \( 1 + (396. - 375. i)T + (6.36e3 - 1.17e5i)T^{2} \) |
| 11 | \( 1 + (129. - 109. i)T + (2.86e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (952. + 2.82e3i)T + (-3.84e6 + 2.92e6i)T^{2} \) |
| 17 | \( 1 + (6.30e3 + 5.97e3i)T + (1.30e6 + 2.41e7i)T^{2} \) |
| 19 | \( 1 + (-2.95e3 - 7.41e3i)T + (-3.41e7 + 3.23e7i)T^{2} \) |
| 23 | \( 1 + (1.02e3 + 9.41e3i)T + (-1.44e8 + 3.18e7i)T^{2} \) |
| 29 | \( 1 + (1.82e4 + 4.01e3i)T + (5.39e8 + 2.49e8i)T^{2} \) |
| 31 | \( 1 + (-1.65e4 - 6.57e3i)T + (6.44e8 + 6.10e8i)T^{2} \) |
| 37 | \( 1 + (-4.36e4 + 5.74e4i)T + (-6.86e8 - 2.47e9i)T^{2} \) |
| 41 | \( 1 + (-3.29e4 - 3.58e3i)T + (4.63e9 + 1.02e9i)T^{2} \) |
| 43 | \( 1 + (3.32e4 + 2.82e4i)T + (1.02e9 + 6.23e9i)T^{2} \) |
| 47 | \( 1 + (-3.03e4 + 8.43e3i)T + (9.23e9 - 5.55e9i)T^{2} \) |
| 53 | \( 1 + (7.12e3 + 1.31e5i)T + (-2.20e10 + 2.39e9i)T^{2} \) |
| 61 | \( 1 + (-5.34e4 - 2.42e5i)T + (-4.67e10 + 2.16e10i)T^{2} \) |
| 67 | \( 1 + (6.60e4 + 8.69e4i)T + (-2.42e10 + 8.71e10i)T^{2} \) |
| 71 | \( 1 + (8.73e4 + 3.14e5i)T + (-1.09e11 + 6.60e10i)T^{2} \) |
| 73 | \( 1 + (-1.96e5 - 1.33e5i)T + (5.60e10 + 1.40e11i)T^{2} \) |
| 79 | \( 1 + (-9.19e3 - 5.60e4i)T + (-2.30e11 + 7.76e10i)T^{2} \) |
| 83 | \( 1 + (-6.68e4 + 3.54e4i)T + (1.83e11 - 2.70e11i)T^{2} \) |
| 89 | \( 1 + (-2.39e5 + 1.08e6i)T + (-4.51e11 - 2.08e11i)T^{2} \) |
| 97 | \( 1 + (2.96e5 - 2.01e5i)T + (3.08e11 - 7.73e11i)T^{2} \) |
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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
−13.32497415317668768887717170968, −12.94019070866074096578823603173, −12.09068205647013868684601639008, −9.214932181881270186590411779505, −8.361795498198403731606286462926, −7.28817848321328124780837141447, −6.15689575340908345779945122596, −5.37525916016264394760498408127, −2.46269839272148690933244813907, −0.33794619387501389664884068506,
2.60604704522752117570455150089, 3.64227754904575546111338967961, 4.45919732065856466355109146390, 6.68488248643407295116062817673, 9.324577760364064822153673786914, 9.885995974788860478302751084746, 10.77582477227486601669281361365, 11.32258764731085213188692794374, 13.27254580734297111853312915809, 14.00061572283069929013595079798
