L(s) = 1 | + (1.10 + 1.91i)2-s + (−1.23 + 2.14i)3-s + (−1.44 + 2.51i)4-s + (−1.06 − 1.83i)5-s − 5.47·6-s + (−2.63 − 0.272i)7-s − 1.98·8-s + (−1.56 − 2.70i)9-s + (2.34 − 4.06i)10-s + (2.39 − 4.14i)11-s + (−3.58 − 6.21i)12-s + (−2.39 − 5.34i)14-s + 5.25·15-s + (0.697 + 1.20i)16-s + (1.88 − 3.27i)17-s + (3.45 − 5.98i)18-s + ⋯ |
L(s) = 1 | + (0.782 + 1.35i)2-s + (−0.714 + 1.23i)3-s + (−0.724 + 1.25i)4-s + (−0.474 − 0.822i)5-s − 2.23·6-s + (−0.994 − 0.102i)7-s − 0.703·8-s + (−0.520 − 0.901i)9-s + (0.742 − 1.28i)10-s + (0.721 − 1.25i)11-s + (−1.03 − 1.79i)12-s + (−0.638 − 1.42i)14-s + 1.35·15-s + (0.174 + 0.301i)16-s + (0.458 − 0.793i)17-s + (0.814 − 1.41i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9113119769\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9113119769\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.63 + 0.272i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.10 - 1.91i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.23 - 2.14i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.06 + 1.83i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.88 + 3.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 + 3.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 + (1.88 - 3.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.81 - 4.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.44 - 2.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + (-3.85 + 6.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.58 - 4.48i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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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
−9.675730217133915048019770454897, −8.844289181046552414267888727175, −8.217619244184841499248278091064, −6.93265087279004216808974993794, −6.32027578696684695382040375672, −5.44350548880500205281357901110, −4.88515280303707583172008215344, −3.98962305337986467696424071924, −3.41244324716963943932487955176, −0.34413024237294033164936519996,
1.42683207716793274764940884928, 2.26397379009176753480227391705, 3.51974463214059652768854227097, 4.08494525304162319241022525522, 5.58573959653048706305825860576, 6.24791979831468884559565892798, 7.19081076619295672696319919050, 7.64684001595271913907805246398, 9.316805510009513251602714750401, 10.05833357760447310923690573268