L(s) = 1 | − 3·5-s + (2.5 − 0.866i)7-s + 3·11-s + (−1 − 1.73i)13-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s − 3·23-s + 4·25-s + (−3 + 5.19i)29-s + (3.5 − 6.06i)31-s + (−7.5 + 2.59i)35-s + (0.5 − 0.866i)37-s + (3 + 5.19i)41-s + (2 − 3.46i)43-s + (−4.5 − 7.79i)47-s + ⋯ |
L(s) = 1 | − 1.34·5-s + (0.944 − 0.327i)7-s + 0.904·11-s + (−0.277 − 0.480i)13-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s − 0.625·23-s + 0.800·25-s + (−0.557 + 0.964i)29-s + (0.628 − 1.08i)31-s + (−1.26 + 0.439i)35-s + (0.0821 − 0.142i)37-s + (0.468 + 0.811i)41-s + (0.304 − 0.528i)43-s + (−0.656 − 1.13i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.453639509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.453639509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.686954904080750992429404424793, −8.046775282463125525462751855021, −7.57573961418871796428557587065, −6.77348592691730488382292906001, −5.72195888197127843599900900255, −4.73478362570569390012341664446, −4.02777428217235854159078374705, −3.38290175680366818695727836974, −1.90531808697282450301918967626, −0.64228351253592272391219834932,
1.02007408875464759529332644698, 2.30405514557169386913691066221, 3.55653432205470019691054163608, 4.26766695122424809613125007561, 4.94948962517530743792586606247, 6.01135986598653580852330347102, 6.98277802225329297155758503026, 7.70688963258776613259040676865, 8.208077386255098953944052598795, 9.026861867605260381224244182765