L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.66 − 0.473i)3-s + (−0.499 + 0.866i)4-s + (0.274 − 0.476i)5-s + (0.423 + 1.67i)6-s + (0.708 + 1.22i)7-s + 0.999·8-s + (2.55 + 1.57i)9-s − 0.549·10-s + (−2.68 − 4.64i)11-s + (1.24 − 1.20i)12-s + (2.46 − 4.27i)13-s + (0.708 − 1.22i)14-s + (−0.683 + 0.663i)15-s + (−0.5 − 0.866i)16-s − 5.10·17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.961 − 0.273i)3-s + (−0.249 + 0.433i)4-s + (0.122 − 0.212i)5-s + (0.172 + 0.685i)6-s + (0.267 + 0.463i)7-s + 0.353·8-s + (0.850 + 0.525i)9-s − 0.173·10-s + (−0.809 − 1.40i)11-s + (0.358 − 0.348i)12-s + (0.684 − 1.18i)13-s + (0.189 − 0.327i)14-s + (−0.176 + 0.171i)15-s + (−0.125 − 0.216i)16-s − 1.23·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0334071 - 0.460430i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0334071 - 0.460430i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (1.66 + 0.473i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.274 + 0.476i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.708 - 1.22i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.68 + 4.64i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.46 + 4.27i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 + (1.63 + 2.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.64 - 4.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 + (-5.82 + 10.0i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.67 + 9.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.98 + 3.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.93 - 8.54i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.36 - 7.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + (-1.72 - 2.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.00 - 3.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 + (7.05 + 12.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.85569582659191786137139661340, −10.33773707021606117650980601183, −8.764413669968963394727187030012, −8.358395129608597188974132700566, −7.02664410751793459582939586684, −5.79053159167471257705595669488, −5.15902627136573332065047741658, −3.61518540696669415574876752478, −2.06673177915636443911724598170, −0.36717894944553653146920656347,
1.86916581949056982266069648125, 4.33874576746438107105843356221, 4.75506322210786798720213129544, 6.30606422758513339200580047479, 6.74147660751211614427823004884, 7.79239829000234865967066248096, 9.023888952109197690162242419965, 9.900515337799107149141312744430, 10.76317691564844776921531920118, 11.27906272706254433989254958176