L(s) = 1 | − 2.50·2-s + (1.71 + 0.207i)3-s + 4.29·4-s − i·5-s + (−4.31 − 0.520i)6-s − i·7-s − 5.75·8-s + (2.91 + 0.713i)9-s + 2.50i·10-s + (3.26 + 0.595i)11-s + (7.38 + 0.890i)12-s − 2.68i·13-s + 2.50i·14-s + (0.207 − 1.71i)15-s + 5.84·16-s + 1.44·17-s + ⋯ |
L(s) = 1 | − 1.77·2-s + (0.992 + 0.119i)3-s + 2.14·4-s − 0.447i·5-s + (−1.76 − 0.212i)6-s − 0.377i·7-s − 2.03·8-s + (0.971 + 0.237i)9-s + 0.793i·10-s + (0.983 + 0.179i)11-s + (2.13 + 0.257i)12-s − 0.744i·13-s + 0.670i·14-s + (0.0535 − 0.443i)15-s + 1.46·16-s + 0.351·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.145674342\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145674342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.71 - 0.207i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-3.26 - 0.595i)T \) |
good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 13 | \( 1 + 2.68iT - 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 - 7.09iT - 19T^{2} \) |
| 23 | \( 1 + 0.250iT - 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 - 1.14iT - 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 11.7iT - 59T^{2} \) |
| 61 | \( 1 - 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 5.66T + 67T^{2} \) |
| 71 | \( 1 - 4.04iT - 71T^{2} \) |
| 73 | \( 1 + 0.248iT - 73T^{2} \) |
| 79 | \( 1 + 14.3iT - 79T^{2} \) |
| 83 | \( 1 + 3.49T + 83T^{2} \) |
| 89 | \( 1 - 11.0iT - 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
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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
−9.697794297056150172387400842933, −8.907037966711842340000706103283, −8.246114624112202311617644669030, −7.69631747203728205802702486728, −6.92823358148386921902783052219, −5.85820074179400204175900936157, −4.26932799931227212045811048194, −3.23982904050223774992983722676, −1.92799472958681413806196994930, −1.02618627916672889099187323600,
1.12606498963434621840932104730, 2.25511632364159099375098064924, 3.07725190676827925186936670634, 4.42604579724940515128931002651, 6.26214550850089093628300029666, 6.87089527146202445857776769781, 7.56994687704450209285671785354, 8.400571741023443878754633586621, 9.233364923767503171432393130062, 9.336644174341597425113742874378