| L(s) = 1 | + (−0.0475 − 1.41i)2-s + (−0.891 + 0.453i)3-s + (−1.99 + 0.134i)4-s + (2.23 − 0.0415i)5-s + (0.684 + 1.23i)6-s + (1.15 − 1.15i)7-s + (0.285 + 2.81i)8-s + (0.587 − 0.809i)9-s + (−0.165 − 3.15i)10-s + (−0.574 − 0.790i)11-s + (1.71 − 1.02i)12-s + (3.66 − 0.580i)13-s + (−1.68 − 1.57i)14-s + (−1.97 + 1.05i)15-s + (3.96 − 0.536i)16-s + (1.54 − 3.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.0336 − 0.999i)2-s + (−0.514 + 0.262i)3-s + (−0.997 + 0.0672i)4-s + (0.999 − 0.0186i)5-s + (0.279 + 0.505i)6-s + (0.437 − 0.437i)7-s + (0.100 + 0.994i)8-s + (0.195 − 0.269i)9-s + (−0.0522 − 0.998i)10-s + (−0.173 − 0.238i)11-s + (0.495 − 0.296i)12-s + (1.01 − 0.161i)13-s + (−0.451 − 0.422i)14-s + (−0.509 + 0.271i)15-s + (0.990 − 0.134i)16-s + (0.374 − 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0730 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0730 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.884691 - 0.822263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.884691 - 0.822263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0475 + 1.41i)T \) |
| 3 | \( 1 + (0.891 - 0.453i)T \) |
| 5 | \( 1 + (-2.23 + 0.0415i)T \) |
| good | 7 | \( 1 + (-1.15 + 1.15i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.574 + 0.790i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 0.580i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 3.02i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.54 + 4.75i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.65 + 0.262i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (6.16 + 2.00i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 1.20i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 7.77i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.58 + 1.87i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.18 - 6.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.67 - 9.17i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.989 + 1.94i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (2.96 + 2.15i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.48 - 2.52i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.00 - 3.05i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (2.89 + 0.939i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.20 + 13.9i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.430 + 1.32i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.43 - 14.5i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.66 - 2.29i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (12.3 - 6.30i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.17809329873089344227695595707, −10.83165589472807392225551221959, −9.787552272574328603342662892670, −9.089500339243648777591249933162, −7.898755726590489570149028425433, −6.30453081281235867264631624965, −5.28447926935050079889160211406, −4.28693196677135442312176553181, −2.78614925122505022606916408332, −1.15801371012226460320328576939,
1.68720898862606681220777556906, 3.99063624000553613150159309185, 5.50372243153198228488597201437, 5.85969584498011925779119690984, 6.92964573721195474460807793575, 8.114834125706900193338027661415, 8.930816122802079054158413613516, 10.03023257587832448140752186685, 10.82610600919512742830128962750, 12.24419373812668738446935060886
