L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 − 2.59i)3-s + (0.500 − 0.866i)4-s + (1.5 + 2.59i)5-s + 3·6-s + (−0.5 − 2.59i)7-s + 3·8-s + (−3 − 5.19i)9-s + (−1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (−1.49 − 2.59i)12-s + (2 − 1.73i)14-s + 9·15-s + (0.500 + 0.866i)16-s + (1 − 1.73i)17-s + (3 − 5.19i)18-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 − 1.49i)3-s + (0.250 − 0.433i)4-s + (0.670 + 1.16i)5-s + 1.22·6-s + (−0.188 − 0.981i)7-s + 1.06·8-s + (−1 − 1.73i)9-s + (−0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + (−0.433 − 0.749i)12-s + (0.534 − 0.462i)14-s + 2.32·15-s + (0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (0.707 − 1.22i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.203383376\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.203383376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 2.59i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−9.915139081518633209634128081390, −8.474630445627107355942521446412, −7.50956611988400964464052029267, −7.10404181606467207098691996136, −6.61751085047204545254630322960, −5.88025568174882864734349845140, −4.54059114014824286517521903492, −3.04104789468140083456004943034, −2.29101666298376808114116792943, −1.21463702303332294204962973448,
1.86504054729845776667317732501, 2.90056379113160664348048663499, 3.53701820624522614986763061884, 4.70989735262425109046900999124, 5.14764001109159023306996383105, 6.21660831718768723493081109880, 7.983749195653116349503547621320, 8.548154181270217259632686312404, 9.012282769476694292121797395843, 9.967938081850056634958906520255