| L(s) = 1 | + (0.309 + 0.951i)2-s + (−2.17 + 1.57i)3-s + (−0.809 + 0.587i)4-s + (1.08 + 1.95i)5-s + (−2.17 − 1.57i)6-s − 3.41·7-s + (−0.809 − 0.587i)8-s + (1.30 − 4.01i)9-s + (−1.52 + 1.63i)10-s + (−0.827 − 2.54i)11-s + (0.830 − 2.55i)12-s + (0.137 − 0.424i)13-s + (−1.05 − 3.24i)14-s + (−5.44 − 2.53i)15-s + (0.309 − 0.951i)16-s + (−0.0132 − 0.00965i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−1.25 + 0.911i)3-s + (−0.404 + 0.293i)4-s + (0.485 + 0.874i)5-s + (−0.887 − 0.644i)6-s − 1.28·7-s + (−0.286 − 0.207i)8-s + (0.434 − 1.33i)9-s + (−0.481 + 0.517i)10-s + (−0.249 − 0.768i)11-s + (0.239 − 0.737i)12-s + (0.0382 − 0.117i)13-s + (−0.281 − 0.866i)14-s + (−1.40 − 0.654i)15-s + (0.0772 − 0.237i)16-s + (−0.00322 − 0.00234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.212769 - 0.0537073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212769 - 0.0537073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-1.08 - 1.95i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| good | 3 | \( 1 + (2.17 - 1.57i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + (0.827 + 2.54i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.137 + 0.424i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.0132 + 0.00965i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + (0.279 + 0.860i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.62 + 4.08i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.32 + 4.59i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.32 - 7.17i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.87 - 5.75i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 + (3.37 - 2.45i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.123 + 0.0893i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.32 + 10.2i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.297 - 0.915i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.00 - 3.63i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-6.92 + 5.03i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.00 + 12.3i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.13 - 0.822i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.81 + 1.32i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.25 + 10.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (4.77 - 3.46i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.873836505584936402854744140965, −9.564920535473241630057799173399, −8.263660489150220568753235684560, −6.99179969030470457719228654718, −6.22113188203346251497556997516, −5.93459301783194508139743157546, −4.92221355403173300846780860996, −3.79057170551191586492991339675, −2.90727541250865439671391339909, −0.12326869715971798959572088071,
1.18494779092537422646602211190, 2.32551983344895799785389616531, 3.86339323326300586338795533435, 5.10441754082034479888947450309, 5.65099162448354194138636676168, 6.58196160733447894087210861575, 7.24602654871696603365703634877, 8.643168418508973648871530759011, 9.479783078129032325082106618735, 10.24962802642471545305773024640
