| L(s) = 1 | + (0.786 + 0.618i)2-s + (0.0372 − 0.0522i)3-s + (0.235 + 0.971i)4-s + (−3.00 + 0.578i)5-s + (0.0615 − 0.0180i)6-s + (−1.64 + 2.06i)7-s + (−0.415 + 0.909i)8-s + (0.979 + 2.83i)9-s + (−2.71 − 1.40i)10-s + (2.48 − 1.95i)11-s + (0.0595 + 0.0238i)12-s + (−4.70 + 3.02i)13-s + (−2.57 + 0.607i)14-s + (−0.0814 + 0.178i)15-s + (−0.888 + 0.458i)16-s + (1.16 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.555 + 0.437i)2-s + (0.0214 − 0.0301i)3-s + (0.117 + 0.485i)4-s + (−1.34 + 0.258i)5-s + (0.0251 − 0.00737i)6-s + (−0.623 + 0.782i)7-s + (−0.146 + 0.321i)8-s + (0.326 + 0.943i)9-s + (−0.859 − 0.442i)10-s + (0.748 − 0.588i)11-s + (0.0171 + 0.00688i)12-s + (−1.30 + 0.839i)13-s + (−0.688 + 0.162i)14-s + (−0.0210 + 0.0460i)15-s + (−0.222 + 0.114i)16-s + (0.283 + 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472936 + 1.03859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472936 + 1.03859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.786 - 0.618i)T \) |
| 7 | \( 1 + (1.64 - 2.06i)T \) |
| 23 | \( 1 + (2.76 - 3.91i)T \) |
| good | 3 | \( 1 + (-0.0372 + 0.0522i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (3.00 - 0.578i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-2.48 + 1.95i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (4.70 - 3.02i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 1.11i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.31 + 1.25i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-9.42 + 2.76i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.51 + 0.335i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (2.66 + 7.68i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-2.15 - 2.49i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 2.23i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (1.76 - 3.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.319 - 6.71i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-12.9 - 6.69i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (3.78 + 5.30i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.672i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.53 - 10.6i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.942 + 3.88i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.394 + 8.28i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-3.86 + 4.45i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.59 + 0.725i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-5.97 - 6.89i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.88107054183403589410267293941, −11.52830211139612037844490909485, −10.11784428659093000156773869904, −8.963858884420561910096287863865, −7.920622500594804722948511068260, −7.19938499288715510779333954665, −6.16752009359604667587695002311, −4.84856902986520839104214733291, −3.87991611762113358778832900288, −2.62524260034067954513469520878,
0.69459364626446054770527274865, 3.10033461790216182703483693427, 4.01914643205808603886947964648, 4.83162642339314249631574538847, 6.56754329153848884133219725282, 7.25383632468583631470372071059, 8.397707834292595610823379043076, 9.832597667086827544217524618746, 10.21377143042519591871364003786, 11.67796122702138693543223287015
