L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.169 − 1.72i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (1.09 − 1.33i)6-s + 1.82·7-s + (−0.707 + 0.707i)8-s + (−2.94 + 0.584i)9-s − 1.00·10-s − 5.15·11-s + (1.72 − 0.169i)12-s + (−3.16 − 3.16i)13-s + (1.29 + 1.29i)14-s + (1.33 + 1.09i)15-s − 1.00·16-s + (2.98 − 2.98i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.0979 − 0.995i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.448 − 0.546i)6-s + 0.690·7-s + (−0.250 + 0.250i)8-s + (−0.980 + 0.194i)9-s − 0.316·10-s − 1.55·11-s + (0.497 − 0.0489i)12-s + (−0.878 − 0.878i)13-s + (0.345 + 0.345i)14-s + (0.345 + 0.283i)15-s − 0.250·16-s + (0.724 − 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6887329513\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6887329513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.169 + 1.72i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (-0.159 + 6.08i)T \) |
good | 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 3.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.98 + 2.98i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.92 + 2.92i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.68 - 3.68i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.89 + 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 + (-6.21 + 6.21i)T - 31iT^{2} \) |
| 41 | \( 1 - 0.816T + 41T^{2} \) |
| 43 | \( 1 + (2.69 + 2.69i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.46iT - 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (-0.207 + 0.207i)T - 59iT^{2} \) |
| 61 | \( 1 + (8.45 - 8.45i)T - 61iT^{2} \) |
| 67 | \( 1 + 12.0iT - 67T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + (-5.32 - 5.32i)T + 79iT^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (-6.37 - 6.37i)T + 89iT^{2} \) |
| 97 | \( 1 + (8.32 + 8.32i)T + 97iT^{2} \) |
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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
−9.444512339673856666649565987653, −8.054747300483763976072149070344, −7.78206996098655470004118586921, −7.31586980612590612678108629769, −6.02163815033237457608548274198, −5.42184722529637020476825582261, −4.53443394215755095586377905287, −2.98571134370111978375640854839, −2.28677569235827716116145856341, −0.23752195430549858769722603634,
1.91988019777235181214925083827, 3.09917221334986438438842053652, 4.15702048474160587040583320890, 4.90878199267888811187289812201, 5.41669946964473292243361620230, 6.61203414744769941347764226607, 8.054563289291820132606805661226, 8.407310754759187913140302890137, 9.653864204812292859606673157438, 10.30705752437324434039515369620