L(s) = 1 | + (2.79 − 0.426i)2-s + 5.32·3-s + (7.63 − 2.38i)4-s − 4.56·5-s + (14.8 − 2.27i)6-s − 24.7i·7-s + (20.3 − 9.92i)8-s + 1.39·9-s + (−12.7 + 1.94i)10-s + 27.3·11-s + (40.6 − 12.7i)12-s + 37.1i·13-s + (−10.5 − 69.2i)14-s − 24.3·15-s + (52.6 − 36.4i)16-s + 119. i·17-s + ⋯ |
L(s) = 1 | + (0.988 − 0.150i)2-s + 1.02·3-s + (0.954 − 0.298i)4-s − 0.408·5-s + (1.01 − 0.154i)6-s − 1.33i·7-s + (0.898 − 0.438i)8-s + 0.0517·9-s + (−0.403 + 0.0615i)10-s + 0.749·11-s + (0.978 − 0.305i)12-s + 0.793i·13-s + (−0.201 − 1.32i)14-s − 0.418·15-s + (0.822 − 0.569i)16-s + 1.71i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.56677 - 0.998598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.56677 - 0.998598i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.79 + 0.426i)T \) |
| 31 | \( 1 + (-114. - 129. i)T \) |
good | 3 | \( 1 - 5.32T + 27T^{2} \) |
| 5 | \( 1 + 4.56T + 125T^{2} \) |
| 7 | \( 1 + 24.7iT - 343T^{2} \) |
| 11 | \( 1 - 27.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 119. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 43.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 65.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.2iT - 2.43e4T^{2} \) |
| 37 | \( 1 + 402. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 84.7iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 410. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 551. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 30.9iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 374. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 12.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 718. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 54.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 566.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 247. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 482.T + 9.12e5T^{2} \) |
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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
−13.08766923674065196595380725991, −11.97252222575859972902084690486, −10.91479219590835014888688183082, −9.864511009662495103726611079166, −8.341334526184470777449778322851, −7.34535795067854244004380105526, −6.15892606325726221111853855102, −4.11364006624630212809361400858, −3.68321466078641237047542134996, −1.77128231669435374768716648266,
2.41824107333213650489860314435, 3.34350968258400225040004796627, 4.94093328065085413942227996170, 6.20292928522104385914999474712, 7.64859240743298334640619551281, 8.566843022608548117426935175087, 9.692557097196618677613781409842, 11.60226429619222424383266448989, 11.90167827730243647788110247795, 13.28287893352036092259111047611