| L(s) = 1 | + 0.236·3-s + (−1.55 − 1.61i)5-s − i·7-s − 2.94·9-s + 3.36i·11-s + 1.01·13-s + (−0.366 − 0.380i)15-s − 0.988i·17-s + 2.29i·19-s − 0.236i·21-s + 0.806i·23-s + (−0.187 + 4.99i)25-s − 1.40·27-s + 3.17i·29-s + 1.77·31-s + ⋯ |
| L(s) = 1 | + 0.136·3-s + (−0.693 − 0.720i)5-s − 0.377i·7-s − 0.981·9-s + 1.01i·11-s + 0.281·13-s + (−0.0946 − 0.0982i)15-s − 0.239i·17-s + 0.526i·19-s − 0.0515i·21-s + 0.168i·23-s + (−0.0374 + 0.999i)25-s − 0.270·27-s + 0.589i·29-s + 0.317·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.249556223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249556223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.55 + 1.61i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.236T + 3T^{2} \) |
| 11 | \( 1 - 3.36iT - 11T^{2} \) |
| 13 | \( 1 - 1.01T + 13T^{2} \) |
| 17 | \( 1 + 0.988iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 - 0.806iT - 23T^{2} \) |
| 29 | \( 1 - 3.17iT - 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 9.87iT - 47T^{2} \) |
| 53 | \( 1 - 4.36T + 53T^{2} \) |
| 59 | \( 1 - 2.34iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 6.35T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 13.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 0.147T + 89T^{2} \) |
| 97 | \( 1 + 9.75iT - 97T^{2} \) |
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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
−8.986704724020793004383421726701, −8.335005048164658901545543530053, −7.64524791696794069761692016283, −6.94325247458517385599192238649, −5.84971877845953317044043533632, −5.05677913923590421408607659788, −4.22398997749915482032067967461, −3.45110530982488966571118461111, −2.26356405795024059579748115106, −0.922080956982900654496902994409,
0.55152348388808205210419810290, 2.41234043005749904644876278137, 3.09866294075274804726011372412, 3.90185202283422401809636273827, 5.00066320295273270961724460865, 6.10763765581816150232837439117, 6.38555777952749207439655845063, 7.67648485643395999885973484456, 8.131988931541505997822970088330, 8.870096044256587738059526931330
