| L(s) = 1 | + (0.431 + 1.34i)2-s + (−1.53 − 0.805i)3-s + (−1.62 + 1.16i)4-s + (−0.788 − 0.360i)5-s + (0.422 − 2.41i)6-s + (1.14 − 3.90i)7-s + (−2.26 − 1.68i)8-s + (1.70 + 2.46i)9-s + (0.144 − 1.21i)10-s + (2.16 − 2.49i)11-s + (3.43 − 0.473i)12-s + (3.70 − 1.08i)13-s + (5.74 − 0.142i)14-s + (0.919 + 1.18i)15-s + (1.29 − 3.78i)16-s + (−4.09 + 0.588i)17-s + ⋯ |
| L(s) = 1 | + (0.305 + 0.952i)2-s + (−0.885 − 0.464i)3-s + (−0.813 + 0.581i)4-s + (−0.352 − 0.161i)5-s + (0.172 − 0.985i)6-s + (0.432 − 1.47i)7-s + (−0.802 − 0.596i)8-s + (0.567 + 0.823i)9-s + (0.0456 − 0.385i)10-s + (0.651 − 0.752i)11-s + (0.990 − 0.136i)12-s + (1.02 − 0.301i)13-s + (1.53 − 0.0381i)14-s + (0.237 + 0.306i)15-s + (0.323 − 0.946i)16-s + (−0.992 + 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.909474 - 0.225414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.909474 - 0.225414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.431 - 1.34i)T \) |
| 3 | \( 1 + (1.53 + 0.805i)T \) |
| 23 | \( 1 + (4.22 + 2.26i)T \) |
| good | 5 | \( 1 + (0.788 + 0.360i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.14 + 3.90i)T + (-5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.16 + 2.49i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.70 + 1.08i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (4.09 - 0.588i)T + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 0.283i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-9.56 + 1.37i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.13 + 4.88i)T + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.156 + 0.343i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.36 - 0.624i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.35 - 9.89i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 + (0.996 - 3.39i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-8.73 + 2.56i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.290 + 0.186i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (3.75 - 3.25i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-2.61 - 3.01i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.43 + 9.98i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-3.66 - 12.4i)T + (-66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.05 - 6.67i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.11 + 1.72i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.86 + 6.27i)T + (-63.5 - 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.80437735106216193778432461357, −11.07583004264099165080784231801, −10.02448719261675977019667429479, −8.409960780089518527616757682902, −7.82277251071769590581737220223, −6.66837539322005072000076510883, −6.08231613978174787567917503190, −4.62004328803049828159689129973, −3.89092455538138587622278092651, −0.790817853346413213943645140333,
1.81066282325209428698452071961, 3.57926754471930938204149672977, 4.69366325350532238690742062538, 5.61957304587484382442085668653, 6.66108283533551826913717559723, 8.588405885921242574978782601516, 9.263798382800129690450120113957, 10.29766988121680987971730593290, 11.33906803522185534988904902718, 11.82141416552473737475336439982
