L(s) = 1 | − 1.21i·2-s + 2.33·3-s + 0.512·4-s + i·5-s − 2.84i·6-s + 3.60i·7-s − 3.06i·8-s + 2.43·9-s + 1.21·10-s + 5.37i·11-s + 1.19·12-s + 4.39·14-s + 2.33i·15-s − 2.71·16-s + 1.13·17-s − 2.97i·18-s + ⋯ |
L(s) = 1 | − 0.862i·2-s + 1.34·3-s + 0.256·4-s + 0.447i·5-s − 1.16i·6-s + 1.36i·7-s − 1.08i·8-s + 0.813·9-s + 0.385·10-s + 1.61i·11-s + 0.344·12-s + 1.17·14-s + 0.602i·15-s − 0.678·16-s + 0.274·17-s − 0.701i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.73087 - 0.386280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.73087 - 0.386280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.21iT - 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 7 | \( 1 - 3.60iT - 7T^{2} \) |
| 11 | \( 1 - 5.37iT - 11T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 2.26iT - 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 + 0.0247T + 29T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 + 8.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.73iT - 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 2.58iT - 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 0.171iT - 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 + 6.39iT - 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 4.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 16.1iT - 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−9.938096825308430814717915965261, −9.432758550968883505391684368936, −8.759132713355858665261823262767, −7.59327638682431634149141234363, −7.01093420883031563655414763344, −5.77294393581941144255023233507, −4.38185973073480649636914480301, −3.25563553938612333802451898929, −2.44188132477319053669433571382, −1.96303641288524439592703328617,
1.29782475790917684072192055650, 2.92739637708683345206152690349, 3.66288863821351000791840156308, 4.94282854353087451321275805029, 6.06730466788803620226640269118, 6.99398678400800700349347429721, 7.82166952220656918976190720563, 8.340808594392782520014199487208, 8.974955121062126399782232581097, 10.12433663312037883527611727966