L(s) = 1 | − 2-s + (−0.707 − 0.707i)3-s + 4-s + (1.69 + 1.45i)5-s + (0.707 + 0.707i)6-s + (−0.764 − 0.764i)7-s − 8-s + 1.00i·9-s + (−1.69 − 1.45i)10-s − 4.86i·11-s + (−0.707 − 0.707i)12-s − 4.63·13-s + (0.764 + 0.764i)14-s + (−0.170 − 2.22i)15-s + 16-s + 5.18i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.759 + 0.650i)5-s + (0.288 + 0.288i)6-s + (−0.288 − 0.288i)7-s − 0.353·8-s + 0.333i·9-s + (−0.536 − 0.460i)10-s − 1.46i·11-s + (−0.204 − 0.204i)12-s − 1.28·13-s + (0.204 + 0.204i)14-s + (−0.0441 − 0.575i)15-s + 0.250·16-s + 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05680311191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05680311191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.69 - 1.45i)T \) |
| 37 | \( 1 + (-0.720 - 6.03i)T \) |
good | 7 | \( 1 + (0.764 + 0.764i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.86iT - 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 - 5.18iT - 17T^{2} \) |
| 19 | \( 1 + (1.84 - 1.84i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + (6.46 + 6.46i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.27 + 1.27i)T - 31iT^{2} \) |
| 41 | \( 1 + 3.55iT - 41T^{2} \) |
| 43 | \( 1 - 7.49T + 43T^{2} \) |
| 47 | \( 1 + (8.00 + 8.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.64 - 5.64i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.790 - 0.790i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.24 - 5.24i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.777 - 0.777i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.599T + 71T^{2} \) |
| 73 | \( 1 + (1.07 + 1.07i)T + 73iT^{2} \) |
| 79 | \( 1 + (9.56 - 9.56i)T - 79iT^{2} \) |
| 83 | \( 1 + (11.2 - 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.70 + 1.70i)T + 89iT^{2} \) |
| 97 | \( 1 - 6.03iT - 97T^{2} \) |
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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
−9.668031721673924086343365639597, −8.495217207846090724336755152825, −7.80092953670352415746342762568, −6.88409515027813347142291563801, −6.06749566370261704527139170636, −5.64565706944475976346933111425, −3.94786124113719077103813624346, −2.72993190003027403132719143370, −1.70900757759135815252425444279, −0.03066742547287904068607518462,
1.77625217485563005401484448171, 2.73967373662642623750452445946, 4.50057846815516703744100927708, 5.08200096974788007754280462153, 6.07839855801539089270490498191, 7.04971047140815543403229865828, 7.73313904521956760663964738067, 9.087887609357727053077478936841, 9.494876559816842733846282309981, 9.923350447961982246428684956392