| L(s) = 1 | + (−1.37 − 0.323i)2-s + (−1.44 + 0.957i)3-s + (1.79 + 0.891i)4-s + (2.16 − 0.555i)5-s + (2.29 − 0.851i)6-s + 0.146·7-s + (−2.17 − 1.80i)8-s + (1.16 − 2.76i)9-s + (−3.16 + 0.0633i)10-s + (−0.745 + 0.541i)11-s + (−3.43 + 0.428i)12-s + (0.637 − 0.876i)13-s + (−0.202 − 0.0475i)14-s + (−2.59 + 2.87i)15-s + (2.41 + 3.19i)16-s + (0.898 + 2.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 − 0.228i)2-s + (−0.833 + 0.552i)3-s + (0.895 + 0.445i)4-s + (0.968 − 0.248i)5-s + (0.937 − 0.347i)6-s + 0.0555·7-s + (−0.769 − 0.638i)8-s + (0.388 − 0.921i)9-s + (−0.999 + 0.0200i)10-s + (−0.224 + 0.163i)11-s + (−0.992 + 0.123i)12-s + (0.176 − 0.243i)13-s + (−0.0540 − 0.0127i)14-s + (−0.669 + 0.742i)15-s + (0.602 + 0.797i)16-s + (0.218 + 0.670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.796694 + 0.0491775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.796694 + 0.0491775i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 0.323i)T \) |
| 3 | \( 1 + (1.44 - 0.957i)T \) |
| 5 | \( 1 + (-2.16 + 0.555i)T \) |
| good | 7 | \( 1 - 0.146T + 7T^{2} \) |
| 11 | \( 1 + (0.745 - 0.541i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.637 + 0.876i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.898 - 2.76i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.30 + 1.72i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.697 + 0.959i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-8.42 - 2.73i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.35 + 1.73i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.39 - 3.29i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.60 - 7.71i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + (-7.15 - 2.32i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.73 + 5.34i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.62 + 6.26i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.04 + 2.21i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.55 - 7.86i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.75 - 5.39i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (9.19 + 12.6i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (14.8 + 4.81i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 4.40i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 2.16i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.03 - 1.63i)T + (78.4 + 57.0i)T^{2} \) |
| 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
−11.59226369115340958677340094392, −10.48718301217756817635141363916, −10.05170301167931630713759225526, −9.203656345717734820868237957289, −8.166351727841545704539870722807, −6.76946519180362316526284358295, −5.95737708015549578341628123706, −4.79955870079876496729524008959, −3.03590338844913660437283098211, −1.21422381114411714018024744767,
1.19508229422589628473261049033, 2.62754015675376612749290698030, 5.17303101785607355432927498042, 5.98618145335914760520689007138, 6.87685737262389405746281658345, 7.73474853047162952677888745300, 8.944436606665422452214943584378, 10.02524287490359472461450207114, 10.55474480944021635629468927069, 11.64115478175314845445548413687
