L(s) = 1 | + i·2-s + i·3-s − 4-s + (−0.353 + 2.20i)5-s − 6-s − 3.94i·7-s − i·8-s − 9-s + (−2.20 − 0.353i)10-s + 1.94·11-s − i·12-s − 2.85i·13-s + 3.94·14-s + (−2.20 − 0.353i)15-s + 16-s − 7.07i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.158 + 0.987i)5-s − 0.408·6-s − 1.49i·7-s − 0.353i·8-s − 0.333·9-s + (−0.698 − 0.111i)10-s + 0.587·11-s − 0.288i·12-s − 0.791i·13-s + 1.05·14-s + (−0.570 − 0.0913i)15-s + 0.250·16-s − 1.71i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.149105194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.149105194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.353 - 2.20i)T \) |
| 37 | \( 1 + iT \) |
good | 7 | \( 1 + 3.94iT - 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 2.85iT - 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 4.68iT - 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 + 6.50T + 31T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 1.09iT - 43T^{2} \) |
| 47 | \( 1 + 6.70iT - 47T^{2} \) |
| 53 | \( 1 - 6.94iT - 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 + 1.33iT - 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 5.08iT - 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 9.62iT - 83T^{2} \) |
| 89 | \( 1 + 2.02T + 89T^{2} \) |
| 97 | \( 1 - 1.13iT - 97T^{2} \) |
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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
−9.836228447862045719130458700091, −9.080573042274030986790741378488, −7.908195025247750222406279862550, −7.25061305002139995710564246130, −6.68612355652844182319237979436, −5.63576228324062439523888778374, −4.49164170753902123704893951955, −3.82925535279596623241851776861, −2.79736409827427722647211706887, −0.53149171999576276133195517229,
1.46311790765117242195307524028, 2.15779988152451439268326632561, 3.60292839020152355331872349072, 4.55727870876329019867727407784, 5.69052809352075866698147595331, 6.19605039783627529005018247806, 7.63694195917959736216355303041, 8.519947236982472916111871351630, 9.015671207022849355825219738962, 9.583227406571261331007686807790