L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.25 − 1.84i)5-s + 1.00·6-s + (3.10 − 3.10i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−2.19 + 0.416i)10-s − 3.82·11-s + (−0.707 − 0.707i)12-s + (0.0891 − 0.0891i)13-s − 4.39·14-s + (0.416 + 2.19i)15-s − 1.00·16-s + (−1.83 + 1.83i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.562 − 0.826i)5-s + 0.408·6-s + (1.17 − 1.17i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.694 + 0.131i)10-s − 1.15·11-s + (−0.204 − 0.204i)12-s + (0.0247 − 0.0247i)13-s − 1.17·14-s + (0.107 + 0.567i)15-s − 0.250·16-s + (−0.443 + 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.638727 - 0.821928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.638727 - 0.821928i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.25 + 1.84i)T \) |
| 19 | \( 1 + (-2.70 + 3.41i)T \) |
good | 7 | \( 1 + (-3.10 + 3.10i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + (-0.0891 + 0.0891i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.83 - 1.83i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.58 - 2.58i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 3.60iT - 31T^{2} \) |
| 37 | \( 1 + (7.07 + 7.07i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 + (-7.93 - 7.93i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.463 - 0.463i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.21 + 3.21i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.40T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 + (-8.78 - 8.78i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.66iT - 71T^{2} \) |
| 73 | \( 1 + (3.64 + 3.64i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.82T + 79T^{2} \) |
| 83 | \( 1 + (0.347 + 0.347i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + (-6.88 - 6.88i)T + 97iT^{2} \) |
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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
−10.79108207513226437289058311354, −9.688212807118013359411148065767, −8.895165356637791148957873356964, −7.914302578605021857994860910597, −7.17882599404248973966299517050, −5.54713633429641002168559770229, −4.84376097947893098641149731169, −3.87576823019195466463100181018, −2.10522909792603599699057653833, −0.75226089350219586088090635219,
1.75237792292057775905766191176, 2.76874744480062839816490167832, 5.02308072944675261996671990277, 5.51700443116705209526287353771, 6.49481450744247386481705909123, 7.47995737000204540598714139102, 8.204927305019591687861892808648, 9.121119993164190139881297303902, 10.19696391939890559357741996814, 10.92981239509854817959475677796