L(s) = 1 | + (1.29 − 0.569i)2-s + (0.611 + 1.62i)3-s + (1.35 − 1.47i)4-s + 2.61·5-s + (1.71 + 1.74i)6-s + 0.280i·7-s + (0.907 − 2.67i)8-s + (−2.25 + 1.98i)9-s + (3.38 − 1.49i)10-s − 1.67i·11-s + (3.21 + 1.28i)12-s + 0.357i·13-s + (0.160 + 0.363i)14-s + (1.60 + 4.24i)15-s + (−0.352 − 3.98i)16-s + 0.581i·17-s + ⋯ |
L(s) = 1 | + (0.915 − 0.402i)2-s + (0.353 + 0.935i)3-s + (0.675 − 0.737i)4-s + 1.17·5-s + (0.700 + 0.713i)6-s + 0.106i·7-s + (0.320 − 0.947i)8-s + (−0.750 + 0.660i)9-s + (1.07 − 0.471i)10-s − 0.505i·11-s + (0.928 + 0.371i)12-s + 0.0991i·13-s + (0.0427 + 0.0972i)14-s + (0.413 + 1.09i)15-s + (−0.0880 − 0.996i)16-s + 0.140i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.14212 + 0.0539794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.14212 + 0.0539794i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 + 0.569i)T \) |
| 3 | \( 1 + (-0.611 - 1.62i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 0.280iT - 7T^{2} \) |
| 11 | \( 1 + 1.67iT - 11T^{2} \) |
| 13 | \( 1 - 0.357iT - 13T^{2} \) |
| 17 | \( 1 - 0.581iT - 17T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 5.89iT - 31T^{2} \) |
| 37 | \( 1 - 7.11iT - 37T^{2} \) |
| 41 | \( 1 + 9.79iT - 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 2.28iT - 59T^{2} \) |
| 61 | \( 1 + 0.378iT - 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 + 15.7iT - 79T^{2} \) |
| 83 | \( 1 - 7.51iT - 83T^{2} \) |
| 89 | \( 1 - 5.80iT - 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−10.51825954457416976686790288282, −10.29159616018567237912088608229, −9.227521423985941425602039817307, −8.438114482179890926116973641195, −6.77806015796191995437178631124, −5.87391153229386663150511212670, −5.12657563313994507956066585934, −4.09857200950016957696708502557, −2.96443220002237111161761342556, −1.93548939626632126989874674006,
1.86533692567350579620123816382, 2.67935663160226369486495635509, 4.14645799567495987476182862186, 5.44693524854383955187786377731, 6.24891763646881184704536564762, 6.91155750400158014234472779470, 7.898968372442812661311074987081, 8.796644365010093394583220755909, 9.866645750340959169809308089842, 10.98424414242264827589971577951