| L(s) = 1 | + (−0.455 + 0.890i)2-s + (−0.738 − 0.646i)3-s + (−0.584 − 0.811i)4-s + (−0.829 + 3.92i)5-s + (0.912 − 0.362i)6-s + (−1.00 + 0.642i)7-s + (0.988 − 0.150i)8-s + (−0.268 − 2.01i)9-s + (−3.11 − 2.52i)10-s + (−1.30 + 0.635i)11-s + (−0.0927 + 0.977i)12-s + (−2.55 − 4.17i)13-s + (−0.112 − 1.18i)14-s + (3.15 − 2.36i)15-s + (−0.316 + 0.948i)16-s + (3.85 + 0.145i)17-s + ⋯ |
| L(s) = 1 | + (−0.322 + 0.629i)2-s + (−0.426 − 0.373i)3-s + (−0.292 − 0.405i)4-s + (−0.371 + 1.75i)5-s + (0.372 − 0.148i)6-s + (−0.380 + 0.242i)7-s + (0.349 − 0.0533i)8-s + (−0.0896 − 0.672i)9-s + (−0.986 − 0.799i)10-s + (−0.392 + 0.191i)11-s + (−0.0267 + 0.282i)12-s + (−0.708 − 1.15i)13-s + (−0.0301 − 0.317i)14-s + (0.814 − 0.610i)15-s + (−0.0790 + 0.237i)16-s + (0.934 + 0.0353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0481830 - 0.173790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0481830 - 0.173790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.455 - 0.890i)T \) |
| 167 | \( 1 + (9.05 + 9.21i)T \) |
| good | 3 | \( 1 + (0.738 + 0.646i)T + (0.396 + 2.97i)T^{2} \) |
| 5 | \( 1 + (0.829 - 3.92i)T + (-4.57 - 2.02i)T^{2} \) |
| 7 | \( 1 + (1.00 - 0.642i)T + (2.95 - 6.34i)T^{2} \) |
| 11 | \( 1 + (1.30 - 0.635i)T + (6.76 - 8.67i)T^{2} \) |
| 13 | \( 1 + (2.55 + 4.17i)T + (-5.92 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-3.85 - 0.145i)T + (16.9 + 1.28i)T^{2} \) |
| 19 | \( 1 + (5.94 - 3.18i)T + (10.5 - 15.8i)T^{2} \) |
| 23 | \( 1 + (7.53 + 1.44i)T + (21.3 + 8.49i)T^{2} \) |
| 29 | \( 1 + (1.64 - 0.315i)T + (26.9 - 10.7i)T^{2} \) |
| 31 | \( 1 + (3.41 - 0.389i)T + (30.2 - 6.97i)T^{2} \) |
| 37 | \( 1 + (-0.584 + 4.38i)T + (-35.7 - 9.68i)T^{2} \) |
| 41 | \( 1 + (-11.1 + 2.57i)T + (36.8 - 17.9i)T^{2} \) |
| 43 | \( 1 + (-7.79 - 8.57i)T + (-4.06 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-2.65 + 2.70i)T + (-0.889 - 46.9i)T^{2} \) |
| 53 | \( 1 + (7.08 - 8.40i)T + (-8.98 - 52.2i)T^{2} \) |
| 59 | \( 1 + (4.99 - 0.189i)T + (58.8 - 4.46i)T^{2} \) |
| 61 | \( 1 + (-2.05 - 4.90i)T + (-42.7 + 43.5i)T^{2} \) |
| 67 | \( 1 + (1.30 + 6.17i)T + (-61.2 + 27.0i)T^{2} \) |
| 71 | \( 1 + (-0.462 - 8.13i)T + (-70.5 + 8.04i)T^{2} \) |
| 73 | \( 1 + (4.36 + 13.0i)T + (-58.4 + 43.7i)T^{2} \) |
| 79 | \( 1 + (8.23 - 5.69i)T + (27.7 - 73.9i)T^{2} \) |
| 83 | \( 1 + (-1.06 - 2.07i)T + (-48.5 + 67.3i)T^{2} \) |
| 89 | \( 1 + (-9.61 - 7.21i)T + (24.9 + 85.4i)T^{2} \) |
| 97 | \( 1 + (-6.06 - 0.691i)T + (94.5 + 21.8i)T^{2} \) |
| 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
−12.22580027367210384801100864600, −10.89578985445409444040159581394, −10.36491632689032268859661062651, −9.448951459548675243158602716780, −7.80526334514583215460695583547, −7.48393415420337001708186871213, −6.16893066255054875260600142176, −5.92678628410302746713924547680, −3.88516278283483352050983043250, −2.61653329293640549748719866138,
0.14034989945085765501128140624, 2.01277532607458043637152193023, 4.07102783355752452196804706475, 4.70311378889916924232303011736, 5.77971440069579806520337776686, 7.54651796744299654491743328228, 8.370907661991258319051832042078, 9.302702462386643849603452597003, 10.01383391626254317273580845704, 11.10674012176182591595383771213
