L(s) = 1 | + (0.390 − 2.61i)7-s + (−2.17 − 1.25i)11-s + 2.11·13-s + (3.89 + 2.24i)17-s + (−4.89 + 2.82i)19-s + (−3.99 − 6.91i)23-s − 4.97i·29-s + (6.13 + 3.54i)31-s + (−8.09 + 4.67i)37-s + 6.23·41-s − 9.47i·43-s + (9.04 − 5.22i)47-s + (−6.69 − 2.04i)49-s + (−3.74 + 6.48i)53-s + (−0.188 + 0.325i)59-s + ⋯ |
L(s) = 1 | + (0.147 − 0.989i)7-s + (−0.656 − 0.378i)11-s + 0.585·13-s + (0.944 + 0.545i)17-s + (−1.12 + 0.648i)19-s + (−0.832 − 1.44i)23-s − 0.923i·29-s + (1.10 + 0.636i)31-s + (−1.33 + 0.768i)37-s + 0.973·41-s − 1.44i·43-s + (1.31 − 0.761i)47-s + (−0.956 − 0.292i)49-s + (−0.514 + 0.890i)53-s + (−0.0245 + 0.0424i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9194589758\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9194589758\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.390 + 2.61i)T \) |
good | 11 | \( 1 + (2.17 + 1.25i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + (-3.89 - 2.24i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.89 - 2.82i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.99 + 6.91i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 + (-6.13 - 3.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.09 - 4.67i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 + 9.47iT - 43T^{2} \) |
| 47 | \( 1 + (-9.04 + 5.22i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.74 - 6.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.188 - 0.325i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.41 - 4.85i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.25 - 1.87i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.01iT - 71T^{2} \) |
| 73 | \( 1 + (2.46 - 4.26i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.97 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.29iT - 83T^{2} \) |
| 89 | \( 1 + (8.85 + 15.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.14T + 97T^{2} \) |
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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
−7.85678372555102526849909743925, −7.07665281586117381789776299835, −6.20510286700323563152831132962, −5.81709078329860326856628022659, −4.68717623465145475410639644701, −4.10447285532750124683525059852, −3.38582967825800365061125246760, −2.36371924163156381444372721893, −1.32446866899298863545380633476, −0.23308181527574353921737814078,
1.32518724474789905507954939264, 2.31715469162314795875777920453, 3.00489508507991629784567434929, 3.97287595641913974047232918016, 4.86279130238827325297373219654, 5.53659292242765637409925702905, 6.08008031005781288095754407880, 6.94335522602454633492139743474, 7.79630422428039631819781699362, 8.209306142565654337280809592650