L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.91 − 1.91i)3-s + 1.00i·4-s − 2.70·6-s + (−2.95 + 2.95i)7-s + (0.707 − 0.707i)8-s − 4.33i·9-s − 3.90·11-s + (1.91 + 1.91i)12-s + (−4.27 + 4.27i)13-s + 4.17·14-s − 1.00·16-s + (−3.82 + 3.82i)17-s + (−3.06 + 3.06i)18-s + (0.797 − 4.28i)19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (1.10 − 1.10i)3-s + 0.500i·4-s − 1.10·6-s + (−1.11 + 1.11i)7-s + (0.250 − 0.250i)8-s − 1.44i·9-s − 1.17·11-s + (0.552 + 0.552i)12-s + (−1.18 + 1.18i)13-s + 1.11·14-s − 0.250·16-s + (−0.926 + 0.926i)17-s + (−0.722 + 0.722i)18-s + (0.183 − 0.983i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0529 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0529 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.295764 + 0.280509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.295764 + 0.280509i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.797 + 4.28i)T \) |
good | 3 | \( 1 + (-1.91 + 1.91i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.95 - 2.95i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 + (4.27 - 4.27i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.82 - 3.82i)T - 17iT^{2} \) |
| 23 | \( 1 + (-4.44 - 4.44i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.10T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (1.05 + 1.05i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.48iT - 41T^{2} \) |
| 43 | \( 1 + (6.63 + 6.63i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.35 - 2.35i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.83 + 3.83i)T - 53iT^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-4.04 - 4.04i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.70iT - 71T^{2} \) |
| 73 | \( 1 + (5.04 + 5.04i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + (2.04 + 2.04i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + (3.35 + 3.35i)T + 97iT^{2} \) |
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
−9.902599279153320976355871127561, −9.170938656214815821179332323050, −8.816723976398536334785241769321, −7.78953157705168907990622614206, −7.08521730140609284037881497179, −6.33315964408045542594861665203, −4.93104202079187326545383494914, −3.33754888726345289678013985720, −2.53521761304904209240279962300, −1.96451939390987612454231174430,
0.17479901187361897175907139350, 2.65508470355362434003717393970, 3.30395400484908539616922707328, 4.54563029222578117769518552536, 5.27818484099785747588867495426, 6.71439059930727562972261014394, 7.52659504531732346319458549671, 8.158491396016405559808139379462, 9.171502089628193594988780691697, 9.777559496552656266617331404701