| L(s) = 1 | + (−0.0813 + 0.0143i)2-s + (−2.99 + 0.0828i)3-s + (−3.75 + 1.36i)4-s + (2.22 + 2.65i)5-s + (0.242 − 0.0497i)6-s + (0.0220 + 0.0381i)7-s + (0.571 − 0.330i)8-s + (8.98 − 0.496i)9-s + (−0.219 − 0.183i)10-s − 11.7i·11-s + (11.1 − 4.40i)12-s + (−15.9 − 13.4i)13-s + (−0.00233 − 0.00278i)14-s + (−6.89 − 7.77i)15-s + (12.1 − 10.2i)16-s + (−13.8 − 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0406 + 0.00717i)2-s + (−0.999 + 0.0276i)3-s + (−0.938 + 0.341i)4-s + (0.445 + 0.530i)5-s + (0.0404 − 0.00829i)6-s + (0.00314 + 0.00544i)7-s + (0.0714 − 0.0412i)8-s + (0.998 − 0.0551i)9-s + (−0.0219 − 0.0183i)10-s − 1.06i·11-s + (0.928 − 0.367i)12-s + (−1.22 − 1.03i)13-s + (−0.000166 − 0.000198i)14-s + (−0.459 − 0.518i)15-s + (0.762 − 0.639i)16-s + (−0.815 − 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0286 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0286 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.373309 - 0.384180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373309 - 0.384180i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 - 0.0828i)T \) |
| 19 | \( 1 + (-18.9 - 1.30i)T \) |
| good | 2 | \( 1 + (0.0813 - 0.0143i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.22 - 2.65i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-0.0220 - 0.0381i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 11.7iT - 121T^{2} \) |
| 13 | \( 1 + (15.9 + 13.4i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (13.8 + 16.5i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-4.89 - 13.4i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (3.51 + 9.65i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 23.9T + 961T^{2} \) |
| 37 | \( 1 - 17.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (67.8 - 11.9i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-21.2 - 7.75i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (17.5 + 48.2i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (74.6 + 13.1i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-8.00 + 22.0i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (79.1 + 66.4i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (17.4 - 98.7i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (26.6 - 4.69i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 4.95i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 9.47i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (20.4 - 11.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-45.4 - 124. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (5.95 + 33.7i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.20857452320098605749872818046, −11.34606702547826211395424424937, −10.16657390872418974970530293237, −9.535921420040315674989777525109, −8.081596789827218083642842350063, −6.94618327426407972157529308258, −5.61618159532260830446886962050, −4.79525997242584517374435235932, −3.10070099749036729009490295073, −0.39911903735731635355086833105,
1.57795719226363631939885013836, 4.47895762065617032606893657700, 4.95231961455986753259526466492, 6.25787315005888710895875512129, 7.45855543408704728963370477952, 9.096956895740825454279474498785, 9.716754806512034714906399860217, 10.64393439588750447374960085627, 12.00266182871780005872754741789, 12.68118896195559217408040020192
