L(s) = 1 | − 1.93i·2-s − 1.73·4-s + 5-s + (1 + 2.44i)7-s − 0.517i·8-s − 1.93i·10-s − 5.27i·11-s − 6.69i·13-s + (4.73 − 1.93i)14-s − 4.46·16-s + 3.46·17-s + 6.69i·19-s − 1.73·20-s − 10.1·22-s + 1.41i·23-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 0.866·4-s + 0.447·5-s + (0.377 + 0.925i)7-s − 0.183i·8-s − 0.610i·10-s − 1.59i·11-s − 1.85i·13-s + (1.26 − 0.516i)14-s − 1.11·16-s + 0.840·17-s + 1.53i·19-s − 0.387·20-s − 2.17·22-s + 0.294i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.695319 - 1.26813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.695319 - 1.26813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + 6.69iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.69iT - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 1.79iT - 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 4.52iT - 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 - 3.58iT - 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.41iT - 71T^{2} \) |
| 73 | \( 1 - 6.69iT - 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−11.30526925941803376764422937402, −10.52009648306080777476934828088, −9.833230481028404740538336253824, −8.675618306953813845613530024764, −7.906440888919063439269924225338, −5.99628308600570897415193958069, −5.39477166136413814713265343408, −3.52459565625723560950418041350, −2.77003279710001407088362527445, −1.20480641101168406966098597349,
2.02453488071682227673094325807, 4.36218637575472330331694073788, 4.97165306204623213123493477573, 6.50067855380537693524795530447, 7.02810906694292914505111306980, 7.80977471373673174662320924812, 9.093853249931903491163407254790, 9.788460906420669317609625384432, 11.04048806836554585635943805136, 11.96485618951658701590300572234