| L(s) = 1 | + (−1.73 + 0.0694i)3-s + (0.354 − 2.20i)5-s + (−1.67 + 1.67i)7-s + (2.99 − 0.240i)9-s + (−5.27 − 1.71i)11-s + (−1.00 + 1.96i)13-s + (−0.459 + 3.84i)15-s + (−4.25 − 0.674i)17-s + (−4.59 − 6.31i)19-s + (2.77 − 3.00i)21-s + (1.53 + 3.01i)23-s + (−4.74 − 1.56i)25-s + (−5.15 + 0.623i)27-s + (0.409 + 0.297i)29-s + (2.41 − 1.75i)31-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0400i)3-s + (0.158 − 0.987i)5-s + (−0.631 + 0.631i)7-s + (0.996 − 0.0801i)9-s + (−1.59 − 0.517i)11-s + (−0.277 + 0.545i)13-s + (−0.118 + 0.992i)15-s + (−1.03 − 0.163i)17-s + (−1.05 − 1.44i)19-s + (0.605 − 0.656i)21-s + (0.320 + 0.628i)23-s + (−0.949 − 0.312i)25-s + (−0.992 + 0.120i)27-s + (0.0759 + 0.0552i)29-s + (0.434 − 0.315i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0337599 - 0.233318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0337599 - 0.233318i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 - 0.0694i)T \) |
| 5 | \( 1 + (-0.354 + 2.20i)T \) |
| good | 7 | \( 1 + (1.67 - 1.67i)T - 7iT^{2} \) |
| 11 | \( 1 + (5.27 + 1.71i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.00 - 1.96i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (4.25 + 0.674i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (4.59 + 6.31i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 3.01i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.409 - 0.297i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.41 + 1.75i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (7.68 + 3.91i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-4.48 + 1.45i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.88 - 5.88i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.527 + 3.33i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-5.21 + 0.825i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.17 - 9.76i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.65 + 5.10i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 4.46i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (6.04 - 8.32i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.93 - 0.983i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-10.1 + 13.9i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.93 + 12.2i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.43 + 4.41i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (15.9 - 2.53i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.31833460675760025735700246859, −10.53145782561965889761518291494, −9.380146860601259336724524440355, −8.680244367950442623468149909104, −7.28801324699484131596193028781, −6.15300500631340982017693325153, −5.27243334611469801199427629849, −4.43981233228986057708786203800, −2.40906262775445159260712722073, −0.18004909432302236872595712495,
2.40464303371254330047185949112, 3.98666367014512344886017861147, 5.28912320387681106815373234337, 6.38187535815401579399024319514, 7.08993182222013599605218564390, 8.092927675642472483325140919050, 9.894731967085816333871147609388, 10.51718632836638234545491197371, 10.81165029134243147487380645445, 12.27286482889324574959252939816
