L(s) = 1 | + (0.707 − 0.707i)2-s − i·3-s − 1.00i·4-s + (−1.15 − 1.15i)5-s + (−0.707 − 0.707i)6-s + (2.02 − 1.70i)7-s + (−0.707 − 0.707i)8-s − 9-s − 1.63·10-s + (1.37 + 1.37i)11-s − 1.00·12-s + (−1.50 − 3.27i)13-s + (0.222 − 2.63i)14-s + (−1.15 + 1.15i)15-s − 1.00·16-s − 1.50·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (−0.517 − 0.517i)5-s + (−0.288 − 0.288i)6-s + (0.763 − 0.645i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s − 0.517·10-s + (0.415 + 0.415i)11-s − 0.288·12-s + (−0.416 − 0.909i)13-s + (0.0593 − 0.704i)14-s + (−0.298 + 0.298i)15-s − 0.250·16-s − 0.365·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.520184 - 1.57186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.520184 - 1.57186i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 7 | \( 1 + (-2.02 + 1.70i)T \) |
| 13 | \( 1 + (1.50 + 3.27i)T \) |
good | 5 | \( 1 + (1.15 + 1.15i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.37 - 1.37i)T + 11iT^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + (0.934 + 0.934i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.19iT - 23T^{2} \) |
| 29 | \( 1 + 0.406T + 29T^{2} \) |
| 31 | \( 1 + (2.31 + 2.31i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.257 - 0.257i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.60 - 3.60i)T + 41iT^{2} \) |
| 43 | \( 1 + 2.46iT - 43T^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + (-8.75 + 8.75i)T - 59iT^{2} \) |
| 61 | \( 1 - 11.7iT - 61T^{2} \) |
| 67 | \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.18 - 5.18i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.56 - 2.56i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 + (-4.90 - 4.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.03 + 2.03i)T - 89iT^{2} \) |
| 97 | \( 1 + (-10.2 - 10.2i)T + 97iT^{2} \) |
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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
−10.68622467140158074561747366136, −9.711247391724814154539681275802, −8.556603814133231842011386715264, −7.72972099547139529794540804481, −6.91510659285921809499471005921, −5.59465141684039342000958097985, −4.65709499604539760677481861391, −3.74967385998910336813199788179, −2.21894489055507114865926331312, −0.844475772182072899889591142894,
2.33912737336825666681579191485, 3.68449735809379613504525882092, 4.54277261530459335503779521909, 5.51789818415329465658579081182, 6.57080029172342849100544506355, 7.47266348344192726764469488826, 8.557118030816410349627315445273, 9.110637407856498803725840511089, 10.42606787767373780177415576704, 11.38112681774762719652998171795