L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.0154 − 1.73i)3-s + (−0.809 − 0.587i)4-s + (−2.53 + 0.823i)5-s + (1.64 + 0.549i)6-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s + (−2.99 − 0.0533i)9-s − 2.66i·10-s + (2.99 + 1.41i)11-s + (−1.03 + 1.39i)12-s + (4.96 + 1.61i)13-s + (−0.587 − 0.809i)14-s + (1.38 + 4.40i)15-s + (0.309 + 0.951i)16-s + (2.02 + 6.22i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.00889 − 0.999i)3-s + (−0.404 − 0.293i)4-s + (−1.13 + 0.368i)5-s + (0.670 + 0.224i)6-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s + (−0.999 − 0.0177i)9-s − 0.842i·10-s + (0.904 + 0.426i)11-s + (−0.297 + 0.401i)12-s + (1.37 + 0.447i)13-s + (−0.157 − 0.216i)14-s + (0.358 + 1.13i)15-s + (0.0772 + 0.237i)16-s + (0.490 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.799547 + 0.489707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799547 + 0.489707i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.0154 + 1.73i)T \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-2.99 - 1.41i)T \) |
good | 5 | \( 1 + (2.53 - 0.823i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-4.96 - 1.61i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.02 - 6.22i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.228 - 0.314i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 7.82iT - 23T^{2} \) |
| 29 | \( 1 + (-8.16 - 5.93i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.410 - 1.26i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.989 - 0.718i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.86 - 1.35i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.75iT - 43T^{2} \) |
| 47 | \( 1 + (-4.74 - 6.53i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.70 + 2.17i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.34 - 3.22i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.02 - 2.93i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 7.83T + 67T^{2} \) |
| 71 | \( 1 + (-8.10 + 2.63i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.44 - 4.74i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 0.435i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.943 + 2.90i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 6.89iT - 89T^{2} \) |
| 97 | \( 1 + (1.15 - 3.54i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.27269378744578760558833243303, −10.43747763468478780948308057579, −8.850282556075579491998874647669, −8.439509781289078490617950947023, −7.53230098518757935013136925726, −6.49925382534334301080560439507, −6.17783994977074641563711781598, −4.40928506120147117322961920809, −3.29967139103560587440530261086, −1.35282221878542941862986571915,
0.74372744293207942124090274579, 3.22368757545961840038091159295, 3.78210401940974754655104557185, 4.75343951895659292131773507991, 6.01310202425953673661559496659, 7.56629076947775015410631611546, 8.440235449824239241614107128508, 9.205819556710841113104288306613, 9.995001457889299297670994148096, 11.06124910600110589039652251627