| L(s) = 1 | + (4.69 + 2.23i)3-s + (5.57 + 9.65i)5-s + (7.78 + 16.8i)7-s + (17.0 + 20.9i)9-s + (−0.436 − 0.251i)11-s + 15.9i·13-s + (4.62 + 57.7i)15-s + (41.0 − 71.0i)17-s + (−55.9 + 32.2i)19-s + (−0.948 + 96.2i)21-s + (−159. + 92.0i)23-s + (0.337 − 0.583i)25-s + (33.2 + 136. i)27-s − 289. i·29-s + (125. + 72.5i)31-s + ⋯ |
| L(s) = 1 | + (0.903 + 0.429i)3-s + (0.498 + 0.863i)5-s + (0.420 + 0.907i)7-s + (0.631 + 0.775i)9-s + (−0.0119 − 0.00690i)11-s + 0.339i·13-s + (0.0795 + 0.994i)15-s + (0.585 − 1.01i)17-s + (−0.675 + 0.389i)19-s + (−0.00985 + 0.999i)21-s + (−1.44 + 0.834i)23-s + (0.00269 − 0.00467i)25-s + (0.237 + 0.971i)27-s − 1.85i·29-s + (0.727 + 0.420i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.833214774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833214774\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.69 - 2.23i)T \) |
| 7 | \( 1 + (-7.78 - 16.8i)T \) |
| good | 5 | \( 1 + (-5.57 - 9.65i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (0.436 + 0.251i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 15.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-41.0 + 71.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (55.9 - 32.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (159. - 92.0i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 289. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-125. - 72.5i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-101. - 176. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 52.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (126. + 218. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-588. - 339. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-249. + 432. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-222. + 128. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-44.0 + 76.3i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 717. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (303. + 175. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-480. - 832. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 177.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-38.5 - 66.7i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 400. iT - 9.12e5T^{2} \) |
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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
−11.37622638930583933858280821636, −10.06094766209723410627217697287, −9.751585373624422476392886150919, −8.525158621022412214293350889078, −7.80564609239200768974147856363, −6.54716184636594525835527236571, −5.45111290434003607941223321809, −4.14906832817680684836527871504, −2.81447634751146842031497446956, −2.01087177008326111668196671852,
0.936127890418747282211265353493, 2.02496341942090774456588350745, 3.64915673921845650445594093668, 4.67805684700445984518618018260, 6.05780396840801852509568987914, 7.21174992186181162915377970721, 8.211906424160054151087789790473, 8.760719456632406166578243389910, 9.936317486049331363425384790470, 10.62192365023288287319648976871
