L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.258 + 0.965i)6-s + (−0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.965 + 0.258i)15-s + 1.00·16-s + i·17-s + (0.965 + 0.258i)19-s + (0.258 − 0.965i)21-s + (0.500 − 0.866i)22-s + i·23-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.258 + 0.965i)6-s + (−0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (0.965 − 0.258i)11-s + (−0.500 + 0.866i)14-s + (0.965 + 0.258i)15-s + 1.00·16-s + i·17-s + (0.965 + 0.258i)19-s + (0.258 − 0.965i)21-s + (0.500 − 0.866i)22-s + i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.240302346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240302346\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.965 - 0.258i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 97 | \( 1 + (0.517 + 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−10.07484401905201351289175688225, −9.408013173610821173922495688457, −8.565603232406119711050396906439, −7.66147355821307755230758828316, −6.35720057702163874528058167957, −5.41360962153666374567616009016, −4.72300335513639834811327317837, −3.83575061896005219278943769166, −3.30406957724403686671233588696, −1.62402751970546023174240358176,
1.08709866809646872358259885576, 2.89594505070418331948134393795, 3.87531513871968707446785302475, 4.94371581211301600581684016243, 6.11065844596119734346392268135, 6.64658149058915866825316501038, 6.97812960000007871181808720226, 7.68355233922151571108385941477, 9.251684879068943421773767449104, 9.917674087550959424844142437591