| L(s) = 1 | + (−0.730 − 1.86i)2-s + (−0.0685 − 0.915i)3-s + (−2.93 + 2.72i)4-s + (14.9 − 10.2i)5-s + (−1.65 + 0.796i)6-s + (5.51 + 17.6i)7-s + (7.20 + 3.47i)8-s + (25.8 − 3.89i)9-s + (−29.9 − 20.4i)10-s + (−59.2 − 8.92i)11-s + (2.69 + 2.49i)12-s + (56.5 − 70.8i)13-s + (28.8 − 23.1i)14-s + (−10.3 − 13.0i)15-s + (1.19 − 15.9i)16-s + (−10.6 + 3.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (−0.0131 − 0.176i)3-s + (−0.366 + 0.340i)4-s + (1.34 − 0.913i)5-s + (−0.112 + 0.0541i)6-s + (0.297 + 0.954i)7-s + (0.318 + 0.153i)8-s + (0.957 − 0.144i)9-s + (−0.947 − 0.646i)10-s + (−1.62 − 0.244i)11-s + (0.0647 + 0.0600i)12-s + (1.20 − 1.51i)13-s + (0.551 − 0.442i)14-s + (−0.178 − 0.223i)15-s + (0.0186 − 0.249i)16-s + (−0.151 + 0.0468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.23046 - 1.11094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23046 - 1.11094i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.730 + 1.86i)T \) |
| 7 | \( 1 + (-5.51 - 17.6i)T \) |
| good | 3 | \( 1 + (0.0685 + 0.915i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (-14.9 + 10.2i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (59.2 + 8.92i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-56.5 + 70.8i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (10.6 - 3.28i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-28.7 + 49.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (76.1 + 23.4i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (40.8 - 179. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-47.5 - 82.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-238. - 221. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (193. + 93.3i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-233. + 112. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-133. - 339. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (227. - 211. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-206. - 141. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-133. - 123. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (331. + 573. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (27.2 + 119. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (355. - 907. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (5.22 - 9.05i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-27.3 - 34.2i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (321. - 48.4i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−12.99791425478606709004582195235, −12.49359377211794224569948322215, −10.82037146689563208224161716156, −10.01019203622212341932813797287, −8.886326448447283476870707831449, −7.999923377394955180897018794993, −5.87513980963139268809917888851, −5.01479027969293315098112434407, −2.69353158193584024666883006612, −1.22407621336370730170080630246,
1.88218958508755974547490300810, 4.22066100667114566142861599166, 5.79715152705520180035129910581, 6.87154990712627504530143646922, 7.87211649538275461812956275751, 9.618497988152802110286188815083, 10.22069644726629814769712752126, 11.09992776602862571352553279145, 13.32328790823779216830285026488, 13.61358844049631230224863435315
