| L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.863 + 0.555i)3-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.910i)5-s + (0.146 − 1.01i)6-s + (0.959 − 0.281i)7-s + (0.841 + 0.540i)8-s + (−0.808 + 1.76i)9-s + (0.959 + 0.281i)10-s + (2.43 + 2.80i)11-s + (0.672 + 0.776i)12-s + (−0.602 − 0.176i)13-s + (−0.415 + 0.909i)14-s + (0.864 + 0.555i)15-s + (−0.959 + 0.281i)16-s + (−0.383 + 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.498 + 0.320i)3-s + (−0.0711 − 0.494i)4-s + (−0.185 − 0.406i)5-s + (0.0596 − 0.414i)6-s + (0.362 − 0.106i)7-s + (0.297 + 0.191i)8-s + (−0.269 + 0.589i)9-s + (0.303 + 0.0891i)10-s + (0.732 + 0.845i)11-s + (0.194 + 0.224i)12-s + (−0.167 − 0.0490i)13-s + (−0.111 + 0.243i)14-s + (0.223 + 0.143i)15-s + (−0.239 + 0.0704i)16-s + (−0.0930 + 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451110 + 0.626427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451110 + 0.626427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.450 - 4.77i)T \) |
| good | 3 | \( 1 + (0.863 - 0.555i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (0.415 + 0.910i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-2.43 - 2.80i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.602 + 0.176i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.383 - 2.66i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.863 - 6.00i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.803 - 5.58i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.41 + 4.12i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (4.84 - 10.6i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.800 - 1.75i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-9.73 + 6.25i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 + (3.99 - 1.17i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-9.40 - 2.76i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (5.57 + 3.58i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.45 + 9.75i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (7.10 - 8.20i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.505 + 3.51i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-3.68 - 1.08i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-6.54 + 14.3i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (3.53 - 2.27i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.95 + 8.66i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.83026186705786194376943634907, −10.80057502683817967593129064030, −10.06611026480341096737137464625, −9.044651195350220994283009345875, −8.079419495845061178666253783838, −7.24096917464881696090735087515, −5.95702878654674625553589467150, −5.07022326573468226745589519377, −4.01320566928398726820772002417, −1.68446474671824702850924460725,
0.72448068612386931823851168807, 2.64153131628626345350562538584, 3.92035481148012831943699418047, 5.43627376293722920356996282193, 6.65257309130124505450279794868, 7.42266272130577750772578834487, 8.859017137530580700761124786062, 9.262623651570131617409832494006, 10.86360094361320543041981640473, 11.19029702474789167870966089346
