| L(s) = 1 | + (0.971 + 1.02i)2-s + (0.129 − 0.815i)3-s + (−0.111 + 1.99i)4-s + (−1.25 + 1.84i)5-s + (0.963 − 0.659i)6-s − 1.98i·7-s + (−2.15 + 1.82i)8-s + (2.20 + 0.716i)9-s + (−3.12 + 0.503i)10-s + (−2.03 + 3.99i)11-s + (1.61 + 0.348i)12-s + (1.98 + 3.90i)13-s + (2.03 − 1.92i)14-s + (1.34 + 1.26i)15-s + (−3.97 − 0.444i)16-s + (4.10 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (0.687 + 0.726i)2-s + (0.0745 − 0.470i)3-s + (−0.0556 + 0.998i)4-s + (−0.562 + 0.826i)5-s + (0.393 − 0.269i)6-s − 0.748i·7-s + (−0.763 + 0.645i)8-s + (0.734 + 0.238i)9-s + (−0.987 + 0.159i)10-s + (−0.613 + 1.20i)11-s + (0.466 + 0.100i)12-s + (0.551 + 1.08i)13-s + (0.543 − 0.514i)14-s + (0.347 + 0.326i)15-s + (−0.993 − 0.111i)16-s + (0.995 + 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08719 + 1.39722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08719 + 1.39722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.971 - 1.02i)T \) |
| 5 | \( 1 + (1.25 - 1.84i)T \) |
| good | 3 | \( 1 + (-0.129 + 0.815i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + 1.98iT - 7T^{2} \) |
| 11 | \( 1 + (2.03 - 3.99i)T + (-6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 3.90i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.10 - 2.98i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.61 - 0.413i)T + (18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (-3.61 + 1.17i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.33 + 8.40i)T + (-27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (7.30 + 5.30i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (4.78 - 2.43i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.54 - 1.80i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.41 - 3.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.50 + 1.81i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.60 - 0.729i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-11.1 + 5.69i)T + (34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (0.425 + 0.216i)T + (35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 0.304i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (1.12 + 1.54i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (14.3 - 4.67i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.85 + 5.70i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.67 - 0.266i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (8.49 - 2.75i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.6 + 8.43i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.73336121661120967351426071880, −10.73438182180127263613566956649, −9.809688035750249231009518549156, −8.284854628069361903717716621870, −7.40826021525269707904854195002, −7.07644545481801732969067001486, −6.04885690004424601644767764313, −4.44275696586479511420172017407, −3.87585412586761999647547947949, −2.23106654753991774601847716611,
1.01466410170260638057807612257, 3.03620126379875875514396311300, 3.82115794057464128941546970531, 5.27676372227509948633403772688, 5.50987641698618598722241536498, 7.24406689505552732974121404796, 8.674350227283332500054775127202, 9.117957533520781208761084331999, 10.45038322160639072909045785216, 10.93508738003588234684948553505
