L(s) = 1 | + (−0.831 + 2.55i)2-s + (−0.438 − 1.34i)3-s + (−4.23 − 3.07i)4-s + 0.608·5-s + 3.81·6-s + (1.39 + 1.01i)7-s + (7.04 − 5.11i)8-s + (0.797 − 0.579i)9-s + (−0.505 + 1.55i)10-s + (−1.08 − 0.786i)11-s + (−2.29 + 7.06i)12-s + (−1.13 − 3.49i)13-s + (−3.75 + 2.73i)14-s + (−0.266 − 0.821i)15-s + (4.00 + 12.3i)16-s + (−2.23 + 1.62i)17-s + ⋯ |
L(s) = 1 | + (−0.587 + 1.80i)2-s + (−0.253 − 0.779i)3-s + (−2.11 − 1.53i)4-s + 0.272·5-s + 1.55·6-s + (0.528 + 0.383i)7-s + (2.49 − 1.80i)8-s + (0.265 − 0.193i)9-s + (−0.159 + 0.492i)10-s + (−0.326 − 0.237i)11-s + (−0.662 + 2.04i)12-s + (−0.314 − 0.968i)13-s + (−1.00 + 0.729i)14-s + (−0.0688 − 0.212i)15-s + (1.00 + 3.07i)16-s + (−0.542 + 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.576383 - 0.191660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.576383 - 0.191660i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (0.831 - 2.55i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.438 + 1.34i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 - 0.608T + 5T^{2} \) |
| 7 | \( 1 + (-1.39 - 1.01i)T + (2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (1.08 + 0.786i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.13 + 3.49i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.23 - 1.62i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.793 - 2.44i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.436 - 0.316i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.51 + 7.73i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + (0.0321 - 0.0989i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.928 + 2.85i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (2.07 + 6.39i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.26 + 1.64i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.144 - 0.443i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + (3.84 - 2.79i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.07 - 4.41i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (7.84 - 5.70i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.15 + 15.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (12.3 + 9.00i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.03 - 0.751i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.835481550957362046950261609917, −8.603316243545502670490176295038, −8.149820638111946379432252747350, −7.40739360159239509346537799823, −6.61291218808040080105983341379, −5.83159859184859409735701574875, −5.29312370057289781271167632024, −4.07273756478420922443209136029, −1.87628297791212400545230891004, −0.37094832384168314283848817885,
1.50182692631369926074183869676, 2.44890481302600708722926292523, 3.74603088791178889816227499443, 4.56691135474800104502125426534, 5.08229912329363817900728936408, 6.97791119586438202762520710493, 7.967203838685021605738112797786, 8.984626496918636215218618083117, 9.524938361847310752097256700105, 10.24645280779258647757605689075