L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (2.12 − 0.707i)5-s + (2 − 2i)7-s + (0.707 − 0.707i)8-s + (−2 − 0.999i)10-s + 1.41i·11-s + (2 + 2i)13-s − 2.82·14-s − 1.00·16-s + (1.41 + 1.41i)17-s − 2i·19-s + (0.707 + 2.12i)20-s + (1.00 − 1.00i)22-s + (−0.707 + 0.707i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.948 − 0.316i)5-s + (0.755 − 0.755i)7-s + (0.250 − 0.250i)8-s + (−0.632 − 0.316i)10-s + 0.426i·11-s + (0.554 + 0.554i)13-s − 0.755·14-s − 0.250·16-s + (0.342 + 0.342i)17-s − 0.458i·19-s + (0.158 + 0.474i)20-s + (0.213 − 0.213i)22-s + (−0.147 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.919532370\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.919532370\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.24 - 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.89iT - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + (4.24 - 4.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−9.225297871574397273433173579190, −8.290571755501691061566121618402, −7.67838440540140454235634669467, −6.70545335265232050432737311310, −5.92845713128280211356330817351, −4.75062142324063874432511005493, −4.21506224271752623229825094002, −2.89423762386558048344699162384, −1.79894821708675730006079630822, −1.04714976876206036284720017766,
1.11611056084519867501604190693, 2.20009993664548104936973402815, 3.18091118375877566915758119886, 4.65953437597302473208349577852, 5.54403574042600431274758936893, 5.99701264826176902036716063692, 6.80099567352721731052363182102, 7.88695612735330777511254604697, 8.396706459553503121917994909288, 9.134855483323882277586761168790