L(s) = 1 | + (−0.707 + 0.707i)3-s + (2.23 − 0.0836i)5-s + (2.64 − 0.0627i)7-s − 1.00i·9-s − 3.98·11-s + (0.500 − 0.500i)13-s + (−1.52 + 1.63i)15-s + (−1.67 − 1.67i)17-s + 7.21·19-s + (−1.82 + 1.91i)21-s + (5.16 + 5.16i)23-s + (4.98 − 0.373i)25-s + (0.707 + 0.707i)27-s + 3.65i·29-s − 4.93i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.999 − 0.0373i)5-s + (0.999 − 0.0237i)7-s − 0.333i·9-s − 1.20·11-s + (0.138 − 0.138i)13-s + (−0.392 + 0.423i)15-s + (−0.407 − 0.407i)17-s + 1.65·19-s + (−0.398 + 0.417i)21-s + (1.07 + 1.07i)23-s + (0.997 − 0.0747i)25-s + (0.136 + 0.136i)27-s + 0.678i·29-s − 0.886i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.005890217\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.005890217\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.23 + 0.0836i)T \) |
| 7 | \( 1 + (-2.64 + 0.0627i)T \) |
good | 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-0.500 + 0.500i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.67 + 1.67i)T + 17iT^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + (-5.16 - 5.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 + 4.93iT - 31T^{2} \) |
| 37 | \( 1 + (-0.292 + 0.292i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 + 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.305 - 0.305i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.39 - 5.39i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.11iT - 61T^{2} \) |
| 67 | \( 1 + (0.944 - 0.944i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + (-1.38 + 1.38i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.64iT - 79T^{2} \) |
| 83 | \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + (-7.43 - 7.43i)T + 97iT^{2} \) |
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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
−9.374538021694822477603182709158, −8.825270147664462902440347446904, −7.66566670851780137733963543360, −7.13290865342012085964367680693, −5.79759572386708552602248032767, −5.29522642133135004488439989804, −4.78592384315084018443960756523, −3.36148955979407993695996775412, −2.31346891834334979940297797602, −1.07539600389118560525244872380,
1.06581015204560507918505721862, 2.11787839723481740388908280199, 3.05823121146043875506090694981, 4.80828652148871820137763049238, 5.11624300252636047356916427095, 6.04587906206591642184191227980, 6.86970595524963592843753446309, 7.75171624118850950317835968251, 8.424562072346381531121610857259, 9.329879070251381182000137235499