| L(s) = 1 | + (1.41 − 0.00839i)2-s + (−1.76 − 1.87i)3-s + (1.99 − 0.0237i)4-s + (0.730 + 1.61i)5-s + (−2.50 − 2.64i)6-s + (−2.25 − 1.38i)7-s + (2.82 − 0.0503i)8-s + (−0.237 + 3.62i)9-s + (1.04 + 2.27i)10-s + (−1.86 − 2.99i)11-s + (−3.56 − 3.71i)12-s + (−0.112 − 1.14i)13-s + (−3.19 − 1.94i)14-s + (1.74 − 4.21i)15-s + (3.99 − 0.0949i)16-s + (−3.98 − 0.524i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00593i)2-s + (−1.01 − 1.08i)3-s + (0.999 − 0.0118i)4-s + (0.326 + 0.721i)5-s + (−1.02 − 1.07i)6-s + (−0.851 − 0.524i)7-s + (0.999 − 0.0178i)8-s + (−0.0792 + 1.20i)9-s + (0.331 + 0.719i)10-s + (−0.561 − 0.902i)11-s + (−1.02 − 1.07i)12-s + (−0.0312 − 0.316i)13-s + (−0.854 − 0.519i)14-s + (0.450 − 1.08i)15-s + (0.999 − 0.0237i)16-s + (−0.966 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.645120 - 1.43810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.645120 - 1.43810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.00839i)T \) |
| 7 | \( 1 + (2.25 + 1.38i)T \) |
| good | 3 | \( 1 + (1.76 + 1.87i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.730 - 1.61i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.86 + 2.99i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.112 + 1.14i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (3.98 + 0.524i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (-1.07 + 6.52i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (0.872 + 0.994i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.651 + 1.21i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (2.24 - 0.602i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.849 - 2.25i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (2.41 + 12.1i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-8.77 + 2.66i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (2.40 - 3.13i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-3.72 - 5.98i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (6.41 - 8.94i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (5.02 - 1.17i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-0.540 - 0.576i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (6.73 + 10.0i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (5.88 - 2.90i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-1.66 - 12.6i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-6.27 + 7.64i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (0.224 + 0.662i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (-8.44 + 8.44i)T - 97iT^{2} \) |
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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
−10.44662751470413456894385937479, −8.930221094429439353944368986898, −7.37764716808384347406400407305, −7.09258120280091390532759110523, −6.19846921775942309899204241068, −5.78388027916470945099771638636, −4.60691153859627287946720105065, −3.20376468099935554073958827426, −2.34130339401835022530267783053, −0.57607142841437337868792441716,
1.95494117216621467530188316167, 3.43969696052245027526149528270, 4.44415295989287020231039099216, 5.08068428643220210678344743181, 5.83336069705239050632244604249, 6.48322014721897529488076115310, 7.69691162767504939546313950651, 9.074196040595603120294899444387, 9.860950222794014539675729331861, 10.41219784412507417631409981844
