| L(s) = 1 | + (−0.700 − 3.93i)2-s + (−14.2 + 2.52i)3-s + (−15.0 + 5.51i)4-s + (−7.84 + 6.58i)5-s + (19.9 + 54.5i)6-s + (−31.3 − 18.0i)7-s + (32.2 + 55.2i)8-s + (121. − 44.3i)9-s + (31.4 + 26.2i)10-s + (−20.6 + 11.9i)11-s + (200. − 116. i)12-s + (26.5 − 150. i)13-s + (−49.3 + 136. i)14-s + (95.5 − 113. i)15-s + (195. − 165. i)16-s + (423. + 154. i)17-s + ⋯ |
| L(s) = 1 | + (−0.175 − 0.984i)2-s + (−1.58 + 0.280i)3-s + (−0.938 + 0.344i)4-s + (−0.313 + 0.263i)5-s + (0.553 + 1.51i)6-s + (−0.639 − 0.369i)7-s + (0.504 + 0.863i)8-s + (1.50 − 0.547i)9-s + (0.314 + 0.262i)10-s + (−0.170 + 0.0986i)11-s + (1.39 − 0.810i)12-s + (0.157 − 0.890i)13-s + (−0.251 + 0.694i)14-s + (0.424 − 0.506i)15-s + (0.762 − 0.647i)16-s + (1.46 + 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.539862 - 0.150307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539862 - 0.150307i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.700 + 3.93i)T \) |
| 19 | \( 1 + (360. - 5.91i)T \) |
| good | 3 | \( 1 + (14.2 - 2.52i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (7.84 - 6.58i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (31.3 + 18.0i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (20.6 - 11.9i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-26.5 + 150. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-423. - 154. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-149. + 178. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (242. - 88.3i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-1.23e3 - 714. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.19e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-175. - 997. i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-1.56e3 - 1.86e3i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (76.6 + 210. i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (2.43e3 + 2.04e3i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-1.77e3 + 4.87e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (2.41e3 + 2.02e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.54e3 - 4.23e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (1.43e3 + 1.71e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-239. - 1.36e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-9.95e3 + 1.75e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-9.14e3 - 5.28e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.17e3 + 1.23e4i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-1.33e4 - 4.86e3i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−13.02070247390932777354070120160, −12.42943952037566304037054799634, −11.30401063043520712388597716831, −10.52477229157787348308972744953, −9.816060817533167323674418480983, −7.926889823663359255849468653085, −6.23969552560015423123654009348, −4.91804083940843647189252180646, −3.44826166427176437655943545636, −0.77854916176551413807610183810,
0.63444405010826296469949447804, 4.42540482312281795913072907170, 5.72646795066206918254591200791, 6.48804319360879534744940567778, 7.73972776666623386730205618076, 9.297526641068013381787930387736, 10.48652056323210278602503799039, 11.87364956417822695537652757011, 12.62775969937725457584840062919, 13.84807272464145515906832664521
