| L(s) = 1 | + (−1.15 + 0.817i)2-s + (1.49 + 0.880i)3-s + (0.662 − 1.88i)4-s + (2.23 − 0.103i)5-s + (−2.44 + 0.203i)6-s − 3.19i·7-s + (0.778 + 2.71i)8-s + (1.44 + 2.62i)9-s + (−2.49 + 1.94i)10-s + (1.39 − 4.28i)11-s + (2.65 − 2.23i)12-s + (−0.224 − 0.690i)13-s + (2.61 + 3.69i)14-s + (3.42 + 1.81i)15-s + (−3.12 − 2.50i)16-s + (0.0729 + 0.100i)17-s + ⋯ |
| L(s) = 1 | + (−0.815 + 0.578i)2-s + (0.861 + 0.508i)3-s + (0.331 − 0.943i)4-s + (0.998 − 0.0461i)5-s + (−0.996 + 0.0828i)6-s − 1.20i·7-s + (0.275 + 0.961i)8-s + (0.482 + 0.875i)9-s + (−0.788 + 0.615i)10-s + (0.419 − 1.29i)11-s + (0.765 − 0.643i)12-s + (−0.0622 − 0.191i)13-s + (0.699 + 0.986i)14-s + (0.883 + 0.468i)15-s + (−0.780 − 0.625i)16-s + (0.0176 + 0.0243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33630 + 0.281187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33630 + 0.281187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 - 0.817i)T \) |
| 3 | \( 1 + (-1.49 - 0.880i)T \) |
| 5 | \( 1 + (-2.23 + 0.103i)T \) |
| good | 7 | \( 1 + 3.19iT - 7T^{2} \) |
| 11 | \( 1 + (-1.39 + 4.28i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.224 + 0.690i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.0729 - 0.100i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.06 + 4.22i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.70 - 8.33i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.44 - 6.12i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.154 + 0.213i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.93 - 9.02i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.55 - 0.829i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.07iT - 43T^{2} \) |
| 47 | \( 1 + (1.43 + 1.03i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.63 - 4.99i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.148 - 0.458i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 3.57i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.89 + 3.98i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (7.54 + 5.47i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.40 + 10.4i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.77 - 5.19i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.31 + 3.13i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-9.37 - 3.04i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (4.65 + 3.38i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.20587863980882097740159842200, −10.59729976775947152334554429963, −9.703193408055116764154745014884, −9.063355660265416125856397935768, −8.117690993186689410750691912913, −7.13041299609567787427784479462, −6.04732776084826067621890936500, −4.81400509940764077245904168534, −3.23956988682415523926527648841, −1.49015607848992911467666329054,
1.97351558419373737446016354969, 2.38225931601675024448935418660, 4.12234520288932635389412757309, 6.05007509656252459022196448201, 6.98759422602073999415515984120, 8.177785196355570386009049848329, 8.985805491455482868530786425670, 9.629526691120304976160505774001, 10.39653493259025147921528873205, 11.93364607648581830127591409358
