| L(s) = 1 | + (0.322 + 1.37i)2-s + (0.0437 + 1.73i)3-s + (−1.79 + 0.889i)4-s + (−1.79 + 1.33i)5-s + (−2.36 + 0.619i)6-s − 3.04i·7-s + (−1.80 − 2.17i)8-s + (−2.99 + 0.151i)9-s + (−2.41 − 2.03i)10-s + (0.519 + 1.59i)11-s + (−1.61 − 3.06i)12-s + (−1.88 + 5.80i)13-s + (4.18 − 0.982i)14-s + (−2.38 − 3.04i)15-s + (2.41 − 3.18i)16-s + (−0.841 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.228 + 0.973i)2-s + (0.0252 + 0.999i)3-s + (−0.895 + 0.444i)4-s + (−0.802 + 0.596i)5-s + (−0.967 + 0.252i)6-s − 1.14i·7-s + (−0.637 − 0.770i)8-s + (−0.998 + 0.0504i)9-s + (−0.764 − 0.644i)10-s + (0.156 + 0.482i)11-s + (−0.467 − 0.884i)12-s + (−0.523 + 1.61i)13-s + (1.11 − 0.262i)14-s + (−0.617 − 0.786i)15-s + (0.604 − 0.796i)16-s + (−0.204 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.229229 - 0.779899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229229 - 0.779899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.322 - 1.37i)T \) |
| 3 | \( 1 + (-0.0437 - 1.73i)T \) |
| 5 | \( 1 + (1.79 - 1.33i)T \) |
| good | 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 + (-0.519 - 1.59i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.88 - 5.80i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.841 - 1.15i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 2.95i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.33 - 4.09i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.32 - 1.82i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.80 - 5.23i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.25 - 10.0i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (8.40 + 2.73i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.84iT - 43T^{2} \) |
| 47 | \( 1 + (-5.06 + 3.68i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.69 - 7.84i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.64 + 11.2i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.22 - 1.68i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.331 + 0.240i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.657 - 2.02i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.14 - 11.2i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.79 - 4.21i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.333 - 0.108i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.649 - 0.471i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.15951210719739578283643682757, −11.33828740837355025163307737857, −10.28643145398304539510838430974, −9.415597544802984772801429593159, −8.438074643330498025612601373662, −7.16232453958945695876906809892, −6.79245700286243845518194910195, −5.01622854171885246779882535946, −4.23668658999457431180097835668, −3.40315643250158287414506776556,
0.55857267096426623597273321222, 2.35229693229166826786133093713, 3.47223523034576097777293708120, 5.18225177381911647735464446680, 5.84533125422629151907332137136, 7.60323058618320555097489114200, 8.462687170186797239541508227335, 9.113070227052189663333807181497, 10.52185084050982113407582544381, 11.59639650017989134791253317863
