L(s) = 1 | + (0.599 − 1.28i)2-s − 0.936i·3-s + (−1.28 − 1.53i)4-s + (−1.19 − 0.561i)6-s + (−0.468 − 2.60i)7-s + (−2.73 + 0.719i)8-s + 2.12·9-s − 2.39i·11-s + (−1.43 + 1.19i)12-s − 2·13-s + (−3.61 − 0.961i)14-s + (−0.719 + 3.93i)16-s − 7.12·17-s + (1.27 − 2.71i)18-s + 2.39·19-s + ⋯ |
L(s) = 1 | + (0.424 − 0.905i)2-s − 0.540i·3-s + (−0.640 − 0.768i)4-s + (−0.489 − 0.229i)6-s + (−0.176 − 0.984i)7-s + (−0.967 + 0.254i)8-s + 0.707·9-s − 0.723i·11-s + (−0.415 + 0.346i)12-s − 0.554·13-s + (−0.966 − 0.257i)14-s + (−0.179 + 0.983i)16-s − 1.72·17-s + (0.300 − 0.640i)18-s + 0.550·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.162866 + 1.38423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.162866 + 1.38423i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.599 + 1.28i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.468 + 2.60i)T \) |
good | 3 | \( 1 + 0.936iT - 3T^{2} \) |
| 11 | \( 1 + 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 7.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 6.14iT - 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 + 4.27iT - 79T^{2} \) |
| 83 | \( 1 + 0.936iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−10.15589758944311756076383620766, −9.357245978754541617837065868516, −8.384999774186103575740115507057, −7.12403144735200452251874503516, −6.56349322622928300434423066553, −5.17815346838467287060153705529, −4.28986316344789178810004465683, −3.28319322916400185864592123567, −1.95850143219913165373924943942, −0.63702991383881288457738085974,
2.37973209838572399875740062732, 3.71888985637594663230214351531, 4.79034092333521363743421898282, 5.28701845239548811371471178812, 6.66163650161792468060823674509, 7.12602076815296084260154988597, 8.339876586022393455976913632573, 9.251362429375132659915766134846, 9.631013574901068620642669194690, 10.91073774434407618815272170411