L(s) = 1 | + (0.5 − 0.866i)2-s + (−1.01 + 1.40i)3-s + (−0.499 − 0.866i)4-s + (−0.792 − 2.09i)5-s + (0.707 + 1.58i)6-s + (1.41 − 2.23i)7-s − 0.999·8-s + (−0.936 − 2.85i)9-s + (−2.20 − 0.358i)10-s + (4.05 − 2.34i)11-s + (1.72 + 0.178i)12-s + 1.04·13-s + (−1.22 − 2.34i)14-s + (3.73 + 1.01i)15-s + (−0.5 + 0.866i)16-s + (−2.73 + 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.586 + 0.809i)3-s + (−0.249 − 0.433i)4-s + (−0.354 − 0.935i)5-s + (0.288 + 0.645i)6-s + (0.534 − 0.845i)7-s − 0.353·8-s + (−0.312 − 0.950i)9-s + (−0.697 − 0.113i)10-s + (1.22 − 0.706i)11-s + (0.497 + 0.0514i)12-s + 0.289·13-s + (−0.328 − 0.626i)14-s + (0.965 + 0.261i)15-s + (−0.125 + 0.216i)16-s + (−0.664 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.837630 - 0.763662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837630 - 0.763662i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (1.01 - 1.40i)T \) |
| 5 | \( 1 + (0.792 + 2.09i)T \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 11 | \( 1 + (-4.05 + 2.34i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 + (2.73 - 1.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 + 0.715i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + (5.73 - 3.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.30 - 1.33i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 6.92iT - 43T^{2} \) |
| 47 | \( 1 + (-9.71 - 5.60i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.28 - 9.15i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 7.13i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (-1.75 - 3.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.73 - 9.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.06iT - 83T^{2} \) |
| 89 | \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−12.00561307611542362166932178977, −11.09020118344583202034605785485, −10.61639546660014444503872182748, −9.162792204086609005694961823043, −8.630310921534329304600773391109, −6.79394772655997642137231213986, −5.47431791884623793554366099270, −4.38180179470814372402225627226, −3.74480962321032956981540631181, −1.04449690019133177355864368347,
2.23908115222411409264650455295, 4.11837391188203029278525010807, 5.55481585414839716755843652623, 6.51804621938133689878056846632, 7.27215269532361589507926890620, 8.274123477170581697869821328481, 9.526097633699303435693480681945, 11.22065179636784123463909516342, 11.59471108867548496538818172674, 12.54483348893051916972281502952