| L(s) = 1 | + (−0.532 − 1.31i)2-s + (0.235 + 0.171i)3-s + (−1.43 + 1.39i)4-s + (2.21 − 0.325i)5-s + (0.0990 − 0.400i)6-s + 0.234i·7-s + (2.59 + 1.13i)8-s + (−0.900 − 2.77i)9-s + (−1.60 − 2.72i)10-s + (5.50 + 1.78i)11-s + (−0.577 + 0.0833i)12-s + (−0.580 − 1.78i)13-s + (0.307 − 0.124i)14-s + (0.577 + 0.302i)15-s + (0.109 − 3.99i)16-s + (−2.42 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.376 − 0.926i)2-s + (0.136 + 0.0989i)3-s + (−0.716 + 0.697i)4-s + (0.989 − 0.145i)5-s + (0.0404 − 0.163i)6-s + 0.0887i·7-s + (0.915 + 0.401i)8-s + (−0.300 − 0.924i)9-s + (−0.507 − 0.861i)10-s + (1.65 + 0.538i)11-s + (−0.166 + 0.0240i)12-s + (−0.160 − 0.495i)13-s + (0.0822 − 0.0333i)14-s + (0.149 + 0.0781i)15-s + (0.0274 − 0.999i)16-s + (−0.588 − 0.810i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975779 - 0.616268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975779 - 0.616268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.532 + 1.31i)T \) |
| 5 | \( 1 + (-2.21 + 0.325i)T \) |
| good | 3 | \( 1 + (-0.235 - 0.171i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 0.234iT - 7T^{2} \) |
| 11 | \( 1 + (-5.50 - 1.78i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.580 + 1.78i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.42 + 3.34i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.59 - 2.19i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.32 + 0.754i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.89 - 5.35i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.28 - 3.11i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.624 + 1.92i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.13 - 6.58i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 + (6.18 - 8.51i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.06 + 2.22i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (6.66 - 2.16i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.54 + 0.503i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.01 - 6.54i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-2.21 - 1.60i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (14.3 + 4.67i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.3 + 7.53i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.55 - 1.85i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.51 + 10.8i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.11 + 5.66i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.23447463621637766773871802117, −11.36923242316363705677981606706, −10.16435041624963427772258191717, −9.226167301028582872505847585731, −8.987094872752485748281095477338, −7.25601405172394859773301771525, −5.95730803541332660559019135839, −4.43800057975463867409159636497, −3.09370544963349307306495362456, −1.48981268594688842379708633850,
1.83224576723344947446536010958, 4.16342077332538501894819719647, 5.62471739691798797743939261110, 6.41162252878325332174175670793, 7.47307813694150233765655994325, 8.794332551938742295170453920143, 9.332258150998904600368904768985, 10.48158778735777661328834796553, 11.45703416835769666228222844824, 13.07741983342144393101528734183
