| L(s) = 1 | + (−0.684 + 1.87i)2-s + (2.51 − 0.916i)3-s + (−3.06 − 2.57i)4-s + (−10.4 − 1.84i)5-s + 5.35i·6-s + (−6.29 + 35.6i)7-s + (6.92 − 4.00i)8-s + (−15.1 + 12.7i)9-s + (10.6 − 18.3i)10-s + (15.9 + 27.6i)11-s + (−10.0 − 3.66i)12-s + (2.90 − 3.45i)13-s + (−62.7 − 36.2i)14-s + (−28.0 + 4.94i)15-s + (2.77 + 15.7i)16-s + (−41.9 − 49.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (0.484 − 0.176i)3-s + (−0.383 − 0.321i)4-s + (−0.935 − 0.164i)5-s + 0.364i·6-s + (−0.339 + 1.92i)7-s + (0.306 − 0.176i)8-s + (−0.562 + 0.471i)9-s + (0.335 − 0.581i)10-s + (0.437 + 0.757i)11-s + (−0.242 − 0.0881i)12-s + (0.0619 − 0.0738i)13-s + (−1.19 − 0.691i)14-s + (−0.482 + 0.0850i)15-s + (0.0434 + 0.246i)16-s + (−0.597 − 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.267780 + 0.839173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267780 + 0.839173i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.684 - 1.87i)T \) |
| 37 | \( 1 + (-16.2 - 224. i)T \) |
| good | 3 | \( 1 + (-2.51 + 0.916i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (10.4 + 1.84i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (6.29 - 35.6i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-15.9 - 27.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-2.90 + 3.45i)T + (-381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (41.9 + 49.9i)T + (-853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (3.81 + 10.4i)T + (-5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-60.9 - 35.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-213. + 123. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 71.9iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-224. - 188. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 - 331. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-154. + 267. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-87.9 - 498. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (344. - 60.8i)T + (1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (158. - 188. i)T + (-3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (39.9 - 226. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-748. + 272. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + 539.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-947. - 167. i)T + (4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-884. + 742. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (991. - 174. i)T + (6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-1.50e3 - 869. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.93935193955934845728915827231, −13.56366671413250309250209166121, −12.27664870364558402310568512588, −11.46910119926908142822255729183, −9.480352190349236732281403132997, −8.674505769257153079935310181454, −7.75603058430316796151048085302, −6.24085601574395212300383661333, −4.80432934094681580269832658748, −2.64716150214203036879040822083,
0.58493418529527835514759980531, 3.40278909573054412981167963052, 4.10572548171434486431259690073, 6.73357238351651041209500593500, 7.996953580153361632031847602865, 9.073752132297906934742504137934, 10.52257412942324429730674516220, 11.15213365123914093100318735864, 12.45791484143806708864523027697, 13.74120914046243448636809465070
