| L(s) = 1 | + (−1.58 + 1.16i)2-s + (0.203 − 1.07i)3-s + (0.551 − 1.78i)4-s + (1.69 − 1.46i)5-s + (0.935 + 1.94i)6-s + (−2.28 + 1.33i)7-s + (−0.0837 − 0.239i)8-s + (1.67 + 0.656i)9-s + (−0.966 + 4.29i)10-s + (−0.791 − 2.01i)11-s + (−1.81 − 0.958i)12-s + (−0.742 − 6.58i)13-s + (2.05 − 4.78i)14-s + (−1.23 − 2.11i)15-s + (3.50 + 2.38i)16-s + (0.0422 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.826i)2-s + (0.117 − 0.621i)3-s + (0.275 − 0.893i)4-s + (0.756 − 0.654i)5-s + (0.382 + 0.793i)6-s + (−0.862 + 0.505i)7-s + (−0.0296 − 0.0846i)8-s + (0.557 + 0.218i)9-s + (−0.305 + 1.35i)10-s + (−0.238 − 0.608i)11-s + (−0.523 − 0.276i)12-s + (−0.205 − 1.82i)13-s + (0.548 − 1.27i)14-s + (−0.318 − 0.547i)15-s + (0.876 + 0.597i)16-s + (0.0102 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.690412 - 0.224597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.690412 - 0.224597i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.69 + 1.46i)T \) |
| 7 | \( 1 + (2.28 - 1.33i)T \) |
| good | 2 | \( 1 + (1.58 - 1.16i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.203 + 1.07i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.791 + 2.01i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.742 + 6.58i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.0422 - 1.13i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (2.02 + 3.50i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.83 - 0.181i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-9.97 - 2.27i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (1.41 + 0.817i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.57 - 4.86i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-0.233 + 0.484i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (5.06 + 1.77i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (4.41 + 5.98i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-3.27 - 6.19i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.660 - 8.81i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (2.54 + 8.24i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-1.85 - 6.92i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.195 + 0.855i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (6.71 - 9.10i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (12.8 - 7.41i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.2 - 1.49i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (3.58 - 9.12i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 1.88i)T + 97iT^{2} \) |
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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
−12.38778231586184443883525657755, −10.44299092922511603251016519982, −9.951640958465039476330292071030, −8.738657158224737035774163635849, −8.305407929636594225933677389962, −7.07135174206830022885277667453, −6.24850469725168622108406236477, −5.20622911816447807668091033198, −2.87603060962580974595159634210, −0.872603881957843563388673779038,
1.79910105215147589663537289528, 3.17858577813924000051593295321, 4.60359591521548690900209104132, 6.44491790899161707823619217866, 7.24875825721603452971616694254, 8.902581490970617859282073166487, 9.610827337099451583739024231579, 10.10014757096479924903949844441, 10.76417451196374945610107626671, 11.89022100606642636090694165421
