| L(s) = 1 | + (−3.06 + 2.56i)2-s + (14.2 − 2.52i)3-s + (2.82 − 15.7i)4-s + (−7.84 + 6.58i)5-s + (−37.3 + 44.4i)6-s + (31.3 + 18.0i)7-s + (31.7 + 55.5i)8-s + (121. − 44.3i)9-s + (7.18 − 40.3i)10-s + (20.6 − 11.9i)11-s + (0.718 − 232. i)12-s + (26.5 − 150. i)13-s + (−142. + 24.9i)14-s + (−95.5 + 113. i)15-s + (−240. − 89.0i)16-s + (423. + 154. i)17-s + ⋯ |
| L(s) = 1 | + (−0.767 + 0.641i)2-s + (1.58 − 0.280i)3-s + (0.176 − 0.984i)4-s + (−0.313 + 0.263i)5-s + (−1.03 + 1.23i)6-s + (0.639 + 0.369i)7-s + (0.495 + 0.868i)8-s + (1.50 − 0.547i)9-s + (0.0718 − 0.403i)10-s + (0.170 − 0.0986i)11-s + (0.00499 − 1.61i)12-s + (0.157 − 0.890i)13-s + (−0.727 + 0.127i)14-s + (−0.424 + 0.506i)15-s + (−0.937 − 0.347i)16-s + (1.46 + 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.92304 + 0.488905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92304 + 0.488905i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.06 - 2.56i)T \) |
| 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
−14.34122156207196483555545884934, −13.13449765936772765519248017355, −11.49627160115097055902566495082, −10.03172625309911187345575767747, −9.032699861525955743147130436454, −7.914770320492398637384071574273, −7.53646901070250055937710057130, −5.59343397133416216671077483409, −3.32420927435109086331693925804, −1.57053823374469687204918477349,
1.51972423137069391123102098422, 3.14166259197118259245355499033, 4.32810732840203043821787193056, 7.40254578839306137162833221365, 8.111270966381893013665757029898, 9.173651912356274440752154521902, 9.953161424401094198940908173015, 11.37512222712888205913418490754, 12.49241541095045280856863709881, 13.88862953468105445284441746263
