L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + (1.87 + 1.21i)5-s + (0.707 + 0.707i)6-s + (−2.24 + 2.24i)7-s − i·8-s − 1.00i·9-s + (−1.21 + 1.87i)10-s + 5.65i·11-s + (−0.707 + 0.707i)12-s − 4.81i·13-s + (−2.24 − 2.24i)14-s + (2.18 − 0.466i)15-s + 16-s − 7.58·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.839 + 0.543i)5-s + (0.288 + 0.288i)6-s + (−0.849 + 0.849i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.384 + 0.593i)10-s + 1.70i·11-s + (−0.204 + 0.204i)12-s − 1.33i·13-s + (−0.600 − 0.600i)14-s + (0.564 − 0.120i)15-s + 0.250·16-s − 1.83·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.268058346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268058346\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.87 - 1.21i)T \) |
| 37 | \( 1 + (4.17 + 4.42i)T \) |
good | 7 | \( 1 + (2.24 - 2.24i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.81iT - 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + (3.73 - 3.73i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.23iT - 23T^{2} \) |
| 29 | \( 1 + (-3.26 - 3.26i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.98 + 1.98i)T - 31iT^{2} \) |
| 41 | \( 1 + 2.43iT - 41T^{2} \) |
| 43 | \( 1 + 3.34iT - 43T^{2} \) |
| 47 | \( 1 + (-6.04 + 6.04i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.02 - 8.02i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.84 - 9.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.11 - 1.11i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.91 - 6.91i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 + (2.58 - 2.58i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.85 - 2.85i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.906 - 0.906i)T + 83iT^{2} \) |
| 89 | \( 1 + (-10.1 - 10.1i)T + 89iT^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−10.08656890680444998625535777961, −9.173068574188824493989001944613, −8.692940665934348316444657704683, −7.45635608205045596234250320336, −6.89671770719297969613518897938, −6.09299252459573239029220720617, −5.39909902430970867450714379986, −4.11330194827207953584176088899, −2.77445175493330315301819077418, −1.99350590611982748368159116393,
0.49511322925702352762953258500, 2.09254853153426992705922227659, 3.06767691535122413135975734470, 4.22438703923597791830222622037, 4.75032641639247690980656720932, 6.31885959428174716481698152628, 6.62326980343546412060051882145, 8.375329320972583560186295995025, 8.888030745806656518699374668060, 9.404985708793926414380461137649