| L(s) = 1 | + (−0.195 − 1.40i)2-s + (0.453 + 0.891i)3-s + (−1.92 + 0.547i)4-s + (−2.17 + 0.500i)5-s + (1.15 − 0.810i)6-s + (0.974 + 0.974i)7-s + (1.14 + 2.58i)8-s + (−0.587 + 0.809i)9-s + (1.12 + 2.95i)10-s + (1.28 + 1.77i)11-s + (−1.36 − 1.46i)12-s + (0.471 + 2.97i)13-s + (1.17 − 1.55i)14-s + (−1.43 − 1.71i)15-s + (3.40 − 2.10i)16-s + (−0.167 − 0.0855i)17-s + ⋯ |
| L(s) = 1 | + (−0.138 − 0.990i)2-s + (0.262 + 0.514i)3-s + (−0.961 + 0.273i)4-s + (−0.974 + 0.224i)5-s + (0.473 − 0.330i)6-s + (0.368 + 0.368i)7-s + (0.404 + 0.914i)8-s + (−0.195 + 0.269i)9-s + (0.356 + 0.934i)10-s + (0.387 + 0.533i)11-s + (−0.392 − 0.422i)12-s + (0.130 + 0.826i)13-s + (0.313 − 0.415i)14-s + (−0.370 − 0.442i)15-s + (0.850 − 0.526i)16-s + (−0.0407 − 0.0207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.882341 + 0.321002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882341 + 0.321002i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.195 + 1.40i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (2.17 - 0.500i)T \) |
| good | 7 | \( 1 + (-0.974 - 0.974i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.28 - 1.77i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.471 - 2.97i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.167 + 0.0855i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.61 - 8.06i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.127 - 0.802i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (9.37 + 3.04i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.18 - 1.35i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.96 + 1.10i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.94 + 2.13i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.86 - 1.86i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.26 + 2.68i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.90 + 3.51i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-4.74 - 3.44i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.78 - 2.02i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.24 + 6.37i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (9.69 + 3.15i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.675 - 0.106i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.74 - 11.5i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.86 - 3.49i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-4.99 - 6.87i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.10 - 10.0i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.74274114879847295606742496745, −11.06519126567531681682012371452, −10.00344031327083139574170776739, −9.193361525854528343291536157776, −8.250781093117460972700105725035, −7.37595964660403270261014946598, −5.51595868945977604913689056474, −4.19671955354962979981462110590, −3.59026296320751979407227121365, −1.94424086107789132193488954800,
0.75709274318088031148870010249, 3.39681468323542675409397330896, 4.61133068109689708866246772077, 5.77448606196249464178558364026, 7.13961748453969931738332179243, 7.59533212301440380777031537383, 8.601089099862453186424259604341, 9.275932101237695656173536172450, 10.80617619692422431017209397282, 11.60843287675976873529885583302
