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 + (−1.70 − 2.02i)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 + (−0.645 − 0.763i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72329 - 0.736066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72329 - 0.736066i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.70 + 2.02i)T \) |
| 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
−10.71284471879914618193460025814, −9.740709560448797338329104219592, −8.740910442362260153495842218912, −7.88713778973387851082087507700, −6.91975771574221189436610242911, −6.11850036986800387845959940436, −5.30565838504798061066848397244, −3.98956682735462169885194863483, −2.87892280867587561581902536347, −0.992788141302775723273597128740,
1.94584579654210502405883888482, 3.11655460033266213193052286989, 4.07603848979138256535585555528, 5.32597399816263320480394185514, 6.15036213780739162250771044818, 6.98727493066027164422537432673, 8.602169760946450592296013560362, 9.477555183145647578793137444695, 9.986174451752125712425149740974, 10.90325482061420535254223447583