L(s) = 1 | + (−0.933 − 0.539i)2-s + (−0.918 + 1.46i)3-s + (−0.418 − 0.725i)4-s + (−0.5 + 0.866i)5-s + (1.64 − 0.876i)6-s + 3.05i·8-s + (−1.31 − 2.69i)9-s + (0.933 − 0.539i)10-s + (3.84 − 2.21i)11-s + (1.44 + 0.0513i)12-s + 0.955i·13-s + (−0.812 − 1.52i)15-s + (0.812 − 1.40i)16-s + (0.253 + 0.439i)17-s + (−0.228 + 3.22i)18-s + (−4.41 − 2.54i)19-s + ⋯ |
L(s) = 1 | + (−0.660 − 0.381i)2-s + (−0.530 + 0.847i)3-s + (−0.209 − 0.362i)4-s + (−0.223 + 0.387i)5-s + (0.673 − 0.357i)6-s + 1.08i·8-s + (−0.437 − 0.899i)9-s + (0.295 − 0.170i)10-s + (1.15 − 0.669i)11-s + (0.418 + 0.0148i)12-s + 0.265i·13-s + (−0.209 − 0.394i)15-s + (0.203 − 0.351i)16-s + (0.0615 + 0.106i)17-s + (−0.0539 + 0.760i)18-s + (−1.01 − 0.584i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0279 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0279 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.377281 + 0.387980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.377281 + 0.387980i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.918 - 1.46i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.933 + 0.539i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 2.21i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.955iT - 13T^{2} \) |
| 17 | \( 1 + (-0.253 - 0.439i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.41 + 2.54i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.72 - 2.14i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.89iT - 29T^{2} \) |
| 31 | \( 1 + (5.10 - 2.94i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.76 - 6.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 0.492T + 43T^{2} \) |
| 47 | \( 1 + (3.32 - 5.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.90 - 4.56i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.81 - 10.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.399 + 0.230i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.85 - 3.20i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.90iT - 71T^{2} \) |
| 73 | \( 1 + (-5.46 + 3.15i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.38 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + (-3.57 + 6.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.91iT - 97T^{2} \) |
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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
−10.71699413272442369080518117364, −9.767454441825578215314281767685, −8.992363987574846954071126398596, −8.575561161820332665521613255093, −6.98776810361700315515892285041, −6.14393249371500782700740687886, −5.16095786313237379093885506285, −4.18222678566812367655140261161, −3.06699948982544537778256668275, −1.26194976410060455360475211448,
0.42696189204061383428787543788, 1.91702517570304439268736901962, 3.73078474122419850064371094370, 4.72043036893915750725865763828, 6.04984446450334208380197698699, 6.83831675994584204619328995042, 7.58081742628412400908318326387, 8.343725121869889046197282148837, 9.101268697384912408715333056923, 9.979735648283929160792402996168