L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 1.73·10-s + (2.36 + 4.09i)11-s + (0.5 − 0.866i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 6.46·17-s − 6.19·19-s + (−0.866 − 1.49i)20-s + (−2.36 + 4.09i)22-s + (0.633 − 1.09i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.670i)5-s + (−0.188 − 0.327i)7-s − 0.353·8-s − 0.547·10-s + (0.713 + 1.23i)11-s + (0.138 − 0.240i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.56·17-s − 1.42·19-s + (−0.193 − 0.335i)20-s + (−0.504 + 0.873i)22-s + (0.132 − 0.228i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8994193585\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8994193585\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + (-0.633 + 1.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.09 - 3.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + (-1.26 + 2.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.19 + 10.7i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.09 - 1.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + (4.09 - 7.09i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.69 + 8.13i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.09 - 3.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + (-3.09 - 5.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.633 - 1.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.45829840233189293689984031025, −9.172103799665737207237346291179, −8.642864117833244198216535756904, −7.47554885864647770778863422604, −6.77823557383799948722601038240, −6.46583651900422204461457243042, −4.97344963616542716136697732867, −4.25634080333921945733477776879, −3.35374773134845549697044851615, −2.01298696169479382691030566272,
0.33386906735975994085501704648, 1.85237491631129760380828367084, 3.07627098327108403018735489422, 4.19328386521004793904242747645, 4.71519917583657309817246387580, 6.13583817042647730762954402209, 6.45473547018816935600626349402, 8.048792714756181613031127715605, 8.772468840750712344833490555380, 9.197137925833759302187631050129