L(s) = 1 | + (−1.18 + 0.771i)2-s + (1.57 − 0.713i)3-s + (0.808 − 1.82i)4-s + (−1.31 + 2.06i)6-s + (−1.44 + 1.44i)7-s + (0.454 + 2.79i)8-s + (1.98 − 2.25i)9-s + 0.641·11-s + (−0.0287 − 3.46i)12-s + (2.03 − 2.03i)13-s + (0.597 − 2.83i)14-s + (−2.69 − 2.95i)16-s + (4.37 + 4.37i)17-s + (−0.612 + 4.19i)18-s + 4.93·19-s + ⋯ |
L(s) = 1 | + (−0.837 + 0.545i)2-s + (0.911 − 0.411i)3-s + (0.404 − 0.914i)4-s + (−0.538 + 0.842i)6-s + (−0.546 + 0.546i)7-s + (0.160 + 0.987i)8-s + (0.660 − 0.750i)9-s + 0.193·11-s + (−0.00829 − 0.999i)12-s + (0.563 − 0.563i)13-s + (0.159 − 0.756i)14-s + (−0.673 − 0.739i)16-s + (1.06 + 1.06i)17-s + (−0.144 + 0.989i)18-s + 1.13·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39177 + 0.0925334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39177 + 0.0925334i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.18 - 0.771i)T \) |
| 3 | \( 1 + (-1.57 + 0.713i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.44 - 1.44i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.641T + 11T^{2} \) |
| 13 | \( 1 + (-2.03 + 2.03i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 + (3.73 - 3.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 9.84iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + (-3.44 - 3.44i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.20iT - 41T^{2} \) |
| 43 | \( 1 + (-4.37 + 4.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.08 + 4.08i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.83 + 3.83i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.50iT - 59T^{2} \) |
| 61 | \( 1 - 9.64iT - 61T^{2} \) |
| 67 | \( 1 + (-2.59 - 2.59i)T + 67iT^{2} \) |
| 71 | \( 1 - 16.7iT - 71T^{2} \) |
| 73 | \( 1 + (8.40 + 8.40i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.31iT - 79T^{2} \) |
| 83 | \( 1 + (-2.20 - 2.20i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \) |
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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
−10.07665287379988426275701244753, −9.812790123149707325108915746062, −8.774143973927661771169122202233, −8.067359461068310895793126400730, −7.43992080862039046879577494562, −6.23688525043274893304021811639, −5.66668718360146504380180765378, −3.80951423908319985685841383081, −2.60479938390886805178520511747, −1.21065128376013908766765251730,
1.28492801071173044615336512936, 2.87728092913176452420585728666, 3.55773444192026099149856318418, 4.70769592131576647752559553917, 6.48962970543992873679873077292, 7.44372002751364011566835682928, 8.112196182782276699529773809386, 9.199780747221243882980639031752, 9.634135343915593656675555479864, 10.39194846210635056255737851016