L(s) = 1 | − i·2-s − 4-s + (−1.38 − 1.75i)5-s + 0.636i·7-s + i·8-s + (−1.75 + 1.38i)10-s − 3.50·11-s + 0.141i·13-s + 0.636·14-s + 16-s − 2.14i·17-s − 19-s + (1.38 + 1.75i)20-s + 3.50i·22-s + 4.91i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.621 − 0.783i)5-s + 0.240i·7-s + 0.353i·8-s + (−0.554 + 0.439i)10-s − 1.05·11-s + 0.0391i·13-s + 0.170·14-s + 0.250·16-s − 0.519i·17-s − 0.229·19-s + (0.310 + 0.391i)20-s + 0.747i·22-s + 1.02i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6788910342\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6788910342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.38 + 1.75i)T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 0.636iT - 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.141iT - 13T^{2} \) |
| 17 | \( 1 + 2.14iT - 17T^{2} \) |
| 23 | \( 1 - 4.91iT - 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 3.27iT - 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 + 2.49iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 8.14iT - 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 8.37iT - 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 + 9.00iT - 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−9.322244088876908594902047505534, −8.809900800181595804446707499643, −7.88784345992867184527334080945, −7.36811582992396403499578502238, −5.93281555973512734940679126665, −5.12066751269161422079203444382, −4.44409850700301929408574125473, −3.40827036480113921532900802360, −2.44976006038993097396497868064, −1.11747570046908030427770617744,
0.29177037184855514264854367761, 2.32088951410355913559779022034, 3.40740267045391422776335282650, 4.29972417363840354773689967441, 5.21028534647412252353111496706, 6.20525382471038788800965796905, 6.89065747210411967377976573852, 7.67241201912565368196418745885, 8.215248981284183430401736596969, 9.031576973199655752478889772822