| L(s) = 1 | + (−1.41 + 0.100i)2-s + (0.472 − 3.28i)3-s + (1.97 − 0.282i)4-s + (−0.959 + 0.281i)5-s + (−0.337 + 4.67i)6-s + (−2.50 − 2.88i)7-s + (−2.76 + 0.597i)8-s + (−7.68 − 2.25i)9-s + (1.32 − 0.493i)10-s + (3.41 − 5.30i)11-s + (0.00647 − 6.63i)12-s + (−2.21 − 1.91i)13-s + (3.81 + 3.82i)14-s + (0.472 + 3.28i)15-s + (3.84 − 1.11i)16-s + (2.31 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0708i)2-s + (0.272 − 1.89i)3-s + (0.989 − 0.141i)4-s + (−0.429 + 0.125i)5-s + (−0.137 + 1.91i)6-s + (−0.945 − 1.09i)7-s + (−0.977 + 0.211i)8-s + (−2.56 − 0.751i)9-s + (0.419 − 0.156i)10-s + (1.02 − 1.60i)11-s + (0.00187 − 1.91i)12-s + (−0.613 − 0.531i)13-s + (1.02 + 1.02i)14-s + (0.121 + 0.847i)15-s + (0.960 − 0.279i)16-s + (0.560 − 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282599 + 0.624482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282599 + 0.624482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.100i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.07 + 2.53i)T \) |
| good | 3 | \( 1 + (-0.472 + 3.28i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (2.50 + 2.88i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.41 + 5.30i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.21 + 1.91i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.31 + 1.05i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-4.58 - 2.09i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.05i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-7.20 + 1.03i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.44 - 1.30i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (1.04 - 0.306i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-2.20 - 0.316i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 3.74iT - 47T^{2} \) |
| 53 | \( 1 + (-2.63 - 3.03i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.58 - 7.59i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.283 - 1.97i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (4.44 + 6.92i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (0.130 + 0.202i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 3.26i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.19 + 7.14i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.562 + 1.91i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (6.65 + 0.957i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (2.22 + 7.58i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−9.361425101697225417491050744393, −8.450139453245924644926972383213, −7.69265034928861366339118097258, −7.31522770368848026113514210784, −6.30058947101644851409005983034, −5.99496007501306639203940815704, −3.45586444924103745953993394931, −2.84830846884308655617027688394, −1.15771994277299968431706783629, −0.49201931970025052550243156527,
2.30472585116721120033226225581, 3.30653099129145841817607968896, 4.22954461670146501954421836288, 5.31122133201626135021973052516, 6.34291761694492241872905358747, 7.47377282853480623080500801837, 8.531567151176556108555870052181, 9.317168118387951469565870326082, 9.733014401059896654412762489976, 9.984268005500915530492534406379
