| L(s) = 1 | + (−0.309 − 0.951i)2-s + (1.71 − 1.24i)3-s + (−0.809 + 0.587i)4-s + (−2.14 − 0.644i)5-s + (−1.71 − 1.24i)6-s − 7-s + (0.809 + 0.587i)8-s + (0.469 − 1.44i)9-s + (0.0490 + 2.23i)10-s + (−1.71 − 5.26i)11-s + (−0.656 + 2.02i)12-s + (0.963 − 2.96i)13-s + (0.309 + 0.951i)14-s + (−4.48 + 1.56i)15-s + (0.309 − 0.951i)16-s + (−2.23 − 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.993 − 0.721i)3-s + (−0.404 + 0.293i)4-s + (−0.957 − 0.288i)5-s + (−0.702 − 0.510i)6-s − 0.377·7-s + (0.286 + 0.207i)8-s + (0.156 − 0.481i)9-s + (0.0155 + 0.706i)10-s + (−0.515 − 1.58i)11-s + (−0.189 + 0.583i)12-s + (0.267 − 0.822i)13-s + (0.0825 + 0.254i)14-s + (−1.15 + 0.404i)15-s + (0.0772 − 0.237i)16-s + (−0.542 − 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227334 - 1.06429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227334 - 1.06429i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.14 + 0.644i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-1.71 + 1.24i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (1.71 + 5.26i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.963 + 2.96i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.23 + 1.62i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.56 + 2.59i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.681 - 2.09i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.93 + 5.04i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.38 - 6.09i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.75 - 5.39i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.729 + 2.24i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + (5.97 - 4.34i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.58 + 5.50i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.42 + 7.46i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.775 + 2.38i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (5.12 + 3.72i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (2.75 - 2.00i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.452 + 1.39i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.51 + 5.46i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (12.1 + 8.86i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.851 - 2.62i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.01 + 5.09i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.14702991263624777083468081815, −10.29769620041320687768049311573, −8.797475518850386241583896629317, −8.445151979393639893717591844197, −7.73386336650514416263962335952, −6.46871995142878124603895035039, −4.83504772133537546060676145873, −3.35446769426285471129075146413, −2.73440251880966496197598693014, −0.73172569485840210894964311647,
2.54926066185725510067979645716, 4.07371075085979816518698926558, 4.52430794428126764327779170625, 6.40799209136845909897390623065, 7.25565317724194244632899618157, 8.281595603499062525990453768806, 8.896108304657995636599495500457, 9.954415251563315108348067558016, 10.54444408995766616092077485822, 11.96103658521446275503084468390
