L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.89 + 1.18i)5-s + 1.00·6-s + (0.705 − 0.705i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−0.498 − 2.17i)10-s + 1.32·11-s + (−0.707 − 0.707i)12-s + (0.741 − 0.741i)13-s − 0.997·14-s + (−2.17 + 0.498i)15-s − 1.00·16-s + (2.17 − 2.17i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.846 + 0.531i)5-s + 0.408·6-s + (0.266 − 0.266i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.157 − 0.689i)10-s + 0.400·11-s + (−0.204 − 0.204i)12-s + (0.205 − 0.205i)13-s − 0.266·14-s + (−0.562 + 0.128i)15-s − 0.250·16-s + (0.527 − 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20672 + 0.145138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20672 + 0.145138i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.89 - 1.18i)T \) |
| 19 | \( 1 + (0.939 - 4.25i)T \) |
good | 7 | \( 1 + (-0.705 + 0.705i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + (-0.741 + 0.741i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.17 + 2.17i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.08 + 1.08i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 - 5.95iT - 31T^{2} \) |
| 37 | \( 1 + (-1.33 - 1.33i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.531iT - 41T^{2} \) |
| 43 | \( 1 + (-1.53 - 1.53i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.13 + 4.13i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 + (-1.99 - 1.99i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.71iT - 71T^{2} \) |
| 73 | \( 1 + (5.83 + 5.83i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + (9.50 + 9.50i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + (1.11 + 1.11i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41081015805394350650443101349, −10.23932922877603197407667955820, −9.247676152808143703171998793609, −8.319386776078260277956109521874, −7.16120222093711734931729075843, −6.24358739538140913453855509065, −5.22587069186431380453719739303, −3.95387397978695162086052693943, −2.79235058106618836462274465350, −1.31806937097873496580790855478,
1.06443692025271496493354410452, 2.32003951494753541235983530695, 4.37804747427398763969446796852, 5.46711990384104723363427506646, 6.14510760261257280716986907895, 7.02147570529489081061475056325, 8.147408790241143083050414635512, 8.881819196960020879493769309258, 9.703554377550679530851685632073, 10.57255580101063629119879972813