L(s) = 1 | + 0.326i·3-s + (−0.278 − 2.21i)5-s + (2.25 + 1.38i)7-s + 2.89·9-s + 4.91i·11-s − 4.89·13-s + (0.724 − 0.0908i)15-s − 6.14·17-s − 4.72·19-s + (−0.452 + 0.735i)21-s − 6.30·23-s + (−4.84 + 1.23i)25-s + 1.92i·27-s − 4.66·29-s + 4.98·31-s + ⋯ |
L(s) = 1 | + 0.188i·3-s + (−0.124 − 0.992i)5-s + (0.852 + 0.523i)7-s + 0.964·9-s + 1.48i·11-s − 1.35·13-s + (0.187 − 0.0234i)15-s − 1.49·17-s − 1.08·19-s + (−0.0986 + 0.160i)21-s − 1.31·23-s + (−0.969 + 0.247i)25-s + 0.370i·27-s − 0.866·29-s + 0.895·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4514116206\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4514116206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.278 + 2.21i)T \) |
| 7 | \( 1 + (-2.25 - 1.38i)T \) |
good | 3 | \( 1 - 0.326iT - 3T^{2} \) |
| 11 | \( 1 - 4.91iT - 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 + 2.88iT - 37T^{2} \) |
| 41 | \( 1 - 7.38iT - 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 5.09iT - 47T^{2} \) |
| 53 | \( 1 - 1.63iT - 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 8.50iT - 61T^{2} \) |
| 67 | \( 1 - 3.67T + 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 + 0.484iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−9.452549793559519631936367831445, −8.601002457178513553580756553757, −7.85787571428098725062440343017, −7.19785261868697126034036634275, −6.26703182653664746871751053085, −5.00385049020217991997266537503, −4.61852454256312524391217772793, −4.13435213810422371072301448773, −2.14625273788710592249689116543, −1.84837069487193042953933711896,
0.13947791435152056720118461252, 1.82672796515004846745653082650, 2.64400857860347485221349661736, 3.95829550192243277925996902166, 4.43316888281825155316692414918, 5.62672179321194043378354219357, 6.60835970588018991975656572604, 7.05090721534935662630682914405, 7.942618214304178295769948931666, 8.448915141422513527679771114189