| L(s) = 1 | − 0.146i·3-s + i·5-s + 0.146·7-s + 2.97·9-s + 2.68i·11-s − i·13-s + 0.146·15-s + 17-s + 4i·19-s − 0.0214i·21-s + 6.68·23-s + 4·25-s − 0.875i·27-s + 4.39i·29-s − 1.31·31-s + ⋯ |
| L(s) = 1 | − 0.0845i·3-s + 0.447i·5-s + 0.0553·7-s + 0.992·9-s + 0.809i·11-s − 0.277i·13-s + 0.0377·15-s + 0.242·17-s + 0.917i·19-s − 0.00467i·21-s + 1.39·23-s + 0.800·25-s − 0.168i·27-s + 0.815i·29-s − 0.236·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44112 + 0.363938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44112 + 0.363938i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 + 0.146iT - 3T^{2} \) |
| 5 | \( 1 - iT - 5T^{2} \) |
| 7 | \( 1 - 0.146T + 7T^{2} \) |
| 11 | \( 1 - 2.68iT - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 4.39iT - 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 + 6.39T + 41T^{2} \) |
| 43 | \( 1 + 6.83iT - 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 - 8.97iT - 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 8.35iT - 61T^{2} \) |
| 67 | \( 1 - 8.29iT - 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4.29iT - 83T^{2} \) |
| 89 | \( 1 + 5.37T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−11.12276653793539625527456238609, −10.37523705006231048722169346995, −9.632041479135318919574422173640, −8.536197259648979673035450249261, −7.30887483062745941052499499659, −6.89374529792058858785319947810, −5.48729293876702880248313596952, −4.40365126446819029693172904626, −3.15509454325742666081778025482, −1.59390378706072359001261323539,
1.17385737226793396951775005669, 2.97781903036639347855602982805, 4.34343408366506131254024618819, 5.20445279278893314176216270752, 6.50808773797814177736396149671, 7.37942763563082005279096255682, 8.544278330130187197043928253170, 9.250551838115406832604293203414, 10.24627932130071962213567196720, 11.14252568886600841854762147071
