| L(s) = 1 | + (−0.245 − 1.39i)2-s + (2.64 − 1.42i)3-s + (−1.87 + 0.684i)4-s + (6.45 + 1.13i)5-s + (−2.62 − 3.32i)6-s + (0.286 − 6.99i)7-s + (1.41 + 2.44i)8-s + (4.95 − 7.51i)9-s − 9.26i·10-s + (1.45 + 8.25i)11-s + (−3.99 + 4.48i)12-s + (2.04 − 2.44i)13-s + (−9.81 + 1.31i)14-s + (18.6 − 6.17i)15-s + (3.06 − 2.57i)16-s − 0.385i·17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.880 − 0.474i)3-s + (−0.469 + 0.171i)4-s + (1.29 + 0.227i)5-s + (−0.438 − 0.554i)6-s + (0.0409 − 0.999i)7-s + (0.176 + 0.306i)8-s + (0.550 − 0.834i)9-s − 0.926i·10-s + (0.132 + 0.750i)11-s + (−0.332 + 0.373i)12-s + (0.157 − 0.187i)13-s + (−0.700 + 0.0941i)14-s + (1.24 − 0.411i)15-s + (0.191 − 0.160i)16-s − 0.0226i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.78873 - 1.83170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78873 - 1.83170i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.245 + 1.39i)T \) |
| 3 | \( 1 + (-2.64 + 1.42i)T \) |
| 7 | \( 1 + (-0.286 + 6.99i)T \) |
| good | 5 | \( 1 + (-6.45 - 1.13i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 8.25i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 2.44i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + 0.385iT - 289T^{2} \) |
| 19 | \( 1 - 0.0990iT - 361T^{2} \) |
| 23 | \( 1 + (0.767 + 0.644i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-11.6 + 9.80i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-10.6 - 29.3i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-1.67 - 2.90i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-20.1 + 24.0i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (42.3 + 15.4i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (17.9 - 49.4i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (46.9 + 81.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (19.9 - 23.7i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (8.95 - 24.6i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (18.1 - 102. i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-19.6 + 34.0i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (26.3 + 15.2i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (4.59 + 26.0i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-99.3 - 118. i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (44.0 - 121. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−10.59917906020658880805644132414, −9.984093106201530553457575983643, −9.331508749412454462764212011498, −8.249472112278271134925536391840, −7.20602729225588125004387681074, −6.30802230430494032001885416534, −4.73406079648541653533744881162, −3.47357601646331139290405447051, −2.27285685153333164638682954844, −1.25798641868041008467292979514,
1.80559987447016232418457709894, 3.08858778271759127649889882044, 4.67467522907758379247186829955, 5.66384197354808913315610602522, 6.43348065222452805637685412628, 7.928074546452605432700222159876, 8.728686272704149903843146797948, 9.362558249865783626064157802568, 10.00854064855063344796679836679, 11.18104326738908103752026884873
