| L(s) = 1 | + (−0.330 − 0.943i)2-s + (−0.286 + 0.819i)3-s + (−0.781 + 0.623i)4-s + (−0.890 − 2.05i)5-s + 0.868·6-s + (−2.90 − 2.90i)7-s + (0.846 + 0.532i)8-s + (1.75 + 1.40i)9-s + (−1.64 + 1.51i)10-s + (−1.50 + 1.88i)11-s + (−0.286 − 0.819i)12-s + (−2.52 + 4.01i)13-s + (−1.78 + 3.70i)14-s + (1.93 − 0.141i)15-s + (0.222 − 0.974i)16-s + (−1.02 − 1.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.233 − 0.667i)2-s + (−0.165 + 0.473i)3-s + (−0.390 + 0.311i)4-s + (−0.398 − 0.917i)5-s + 0.354·6-s + (−1.09 − 1.09i)7-s + (0.299 + 0.188i)8-s + (0.585 + 0.466i)9-s + (−0.519 + 0.480i)10-s + (−0.453 + 0.568i)11-s + (−0.0828 − 0.236i)12-s + (−0.700 + 1.11i)13-s + (−0.477 + 0.990i)14-s + (0.500 − 0.0365i)15-s + (0.0556 − 0.243i)16-s + (−0.249 − 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0101810 + 0.0229478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0101810 + 0.0229478i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.330 + 0.943i)T \) |
| 5 | \( 1 + (0.890 + 2.05i)T \) |
| 43 | \( 1 + (-2.75 + 5.95i)T \) |
| good | 3 | \( 1 + (0.286 - 0.819i)T + (-2.34 - 1.87i)T^{2} \) |
| 7 | \( 1 + (2.90 + 2.90i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.50 - 1.88i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (2.52 - 4.01i)T + (-5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (1.02 + 1.63i)T + (-7.37 + 15.3i)T^{2} \) |
| 19 | \( 1 + (-2.82 - 3.53i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (7.51 - 0.846i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (6.37 + 3.07i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (7.32 + 3.52i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 3.17i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.02 - 0.492i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (1.03 + 0.116i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-11.5 + 7.27i)T + (22.9 - 47.7i)T^{2} \) |
| 59 | \( 1 + (-5.28 - 1.20i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-3.20 - 6.65i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (13.2 + 1.49i)T + (65.3 + 14.9i)T^{2} \) |
| 71 | \( 1 + (1.89 - 1.51i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.24 + 3.92i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + 4.80iT - 79T^{2} \) |
| 83 | \( 1 + (0.0879 + 0.0307i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (-7.09 + 3.41i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (-1.56 - 13.8i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−11.53993655657465819057675936413, −10.32232508874329185254204996693, −9.808147774893505955845943796221, −9.229262598157298732654253094690, −7.71314204597027255231735392192, −7.22223874084593922347779748295, −5.50703571472566210275663046693, −4.24134714816069898493472452661, −3.90095773742505107086840730208, −1.91976505656733166470819590612,
0.01669403588089155375712271089, 2.57729821222628157764895958305, 3.70935572553517642571224723103, 5.55161057708720658826889673982, 6.15161928580150983858471461493, 7.14367184453322946322671127727, 7.78066015558223608394639921757, 9.001638027674576003901203489925, 9.832461501084778056892977954140, 10.68194710259955049648430976156
