L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 − 0.866i)3-s + (0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.258 + 0.965i)11-s + (−0.500 − 0.866i)14-s + (−0.258 − 0.965i)15-s + 1.00·16-s + i·17-s + (−0.258 − 0.965i)19-s + (−0.965 + 0.258i)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 + (0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.258 + 0.965i)11-s + (−0.500 − 0.866i)14-s + (−0.258 − 0.965i)15-s + 1.00·16-s + i·17-s + (−0.258 − 0.965i)19-s + (−0.965 + 0.258i)21-s + (0.500 + 0.866i)22-s + i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.575824092\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.575824092\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.258 + 0.965i)T \) |
| 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 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 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 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \) |
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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
−10.13544798173690550829078335673, −9.150768607010025768276515032300, −7.76586847540454900724549617511, −7.30927089471544553255988697350, −6.40717620226909456341458865201, −5.51355345027283307116005215748, −4.50243432948906182873191942930, −3.63940421250075811332234338697, −2.22934974392979823974876497369, −1.56319410062699380747270530916,
1.76946707379895979292838663226, 3.24055530891732823347953491505, 4.64761400146674548021442464693, 5.07865768492404828620873706143, 5.87335815210409334268220758046, 6.22349584440960533782044556852, 7.53926784363859690239604208978, 8.615328908406025913453404948398, 9.399983469892542045444680505888, 10.19824238223610739707654278657