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
−9.329879070251381182000137235499, −8.424562072346381531121610857259, −7.75171624118850950317835968251, −6.86970595524963592843753446309, −6.04587906206591642184191227980, −5.11624300252636047356916427095, −4.80828652148871820137763049238, −3.05823121146043875506090694981, −2.11787839723481740388908280199, −1.06581015204560507918505721862,
1.07539600389118560525244872380, 2.31346891834334979940297797602, 3.36148955979407993695996775412, 4.78592384315084018443960756523, 5.29522642133135004488439989804, 5.79759572386708552602248032767, 7.13290865342012085964367680693, 7.66566670851780137733963543360, 8.825270147664462902440347446904, 9.374538021694822477603182709158