L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.292 + 1.70i)3-s − 1.00i·4-s + (0.707 − 2.12i)5-s + (−0.999 − 1.41i)6-s + (−1 − i)7-s + (0.707 + 0.707i)8-s + (−2.82 − i)9-s + (0.999 + 2i)10-s + 1.41i·11-s + (1.70 + 0.292i)12-s + 1.41·14-s + (3.41 + 1.82i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + (2.70 − 1.29i)18-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.169 + 0.985i)3-s − 0.500i·4-s + (0.316 − 0.948i)5-s + (−0.408 − 0.577i)6-s + (−0.377 − 0.377i)7-s + (0.250 + 0.250i)8-s + (−0.942 − 0.333i)9-s + (0.316 + 0.632i)10-s + 0.426i·11-s + (0.492 + 0.0845i)12-s + 0.377·14-s + (0.881 + 0.472i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + (0.638 − 0.304i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.500941 + 0.225904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.500941 + 0.225904i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.292 - 1.70i)T \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (-6 - 6i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (4 + 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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
−16.90375285714291204555643085105, −16.32141304116390009919213512709, −15.19197698537520675261550340977, −13.85289691093411225209009374022, −12.23668530395967147757095350018, −10.45471285693397489650266968309, −9.526070664807416336095550719717, −8.279743490344514498393476746399, −6.10595906160915434301636909297, −4.50368004225802633236355559239,
2.68850278398652390264720053394, 6.16709329434069627483400906865, 7.49688463317062462469160487759, 9.142985922861602836403785620908, 10.76674121723936565168200091958, 11.80062970170223287271475144603, 13.14067675765468224873364350630, 14.20126631705845841631271124206, 15.89890463846466991816436824987, 17.48307595123005864603652837595