L(s) = 1 | + (0.809 − 0.587i)2-s + (0.0373 + 0.115i)3-s + (0.309 − 0.951i)4-s + (1.75 + 1.27i)5-s + (0.0978 + 0.0711i)6-s + (−1.23 + 3.79i)7-s + (−0.309 − 0.951i)8-s + (2.41 − 1.75i)9-s + 2.16·10-s + (3.30 − 0.308i)11-s + 0.120·12-s + (−4.23 + 3.07i)13-s + (1.23 + 3.79i)14-s + (−0.0810 + 0.249i)15-s + (−0.809 − 0.587i)16-s + (0.125 + 0.0911i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.0215 + 0.0664i)3-s + (0.154 − 0.475i)4-s + (0.784 + 0.570i)5-s + (0.0399 + 0.0290i)6-s + (−0.465 + 1.43i)7-s + (−0.109 − 0.336i)8-s + (0.805 − 0.584i)9-s + 0.685·10-s + (0.995 − 0.0929i)11-s + 0.0349·12-s + (−1.17 + 0.853i)13-s + (0.329 + 1.01i)14-s + (−0.0209 + 0.0644i)15-s + (−0.202 − 0.146i)16-s + (0.0304 + 0.0221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12759 + 0.0737681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12759 + 0.0737681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-3.30 + 0.308i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.0373 - 0.115i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.75 - 1.27i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (1.23 - 3.79i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (4.23 - 3.07i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.125 - 0.0911i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 + (-1.66 + 5.12i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.64 - 1.92i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.17 + 6.70i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.28 + 10.1i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 9.72T + 43T^{2} \) |
| 47 | \( 1 + (-2.05 - 6.32i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (4.73 - 3.43i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.34 - 4.13i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.21 - 1.60i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 + (2.97 + 2.16i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.06 - 3.26i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.416 + 0.302i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (11.2 + 8.14i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 + (9.81 - 7.12i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.44187101446984935249109933616, −10.21611700150881480192287722587, −9.422834496574666659246809338413, −9.016486970454131668060656861364, −7.03053566347314426500967981644, −6.44101507540255065976060474447, −5.48401741667663329844558766926, −4.26102898875002286166370510471, −2.91497155044441044739134445790, −1.92842965778385797782818787098,
1.42034559725598926367097624289, 3.27332816963493076962960294088, 4.53155836175978671362406679447, 5.23061951344185429203433393660, 6.68745546959956763985926472135, 7.16906153328475438299120022868, 8.247791629738322051145692450714, 9.654145325804138703184913830122, 10.06649699985523724182863116632, 11.19530400073357544661441049707