| L(s) = 1 | + (0.811 − 1.15i)2-s + (1.70 + 0.300i)3-s + (−0.684 − 1.87i)4-s + (−9.26 + 0.810i)5-s + (1.73 − 1.73i)6-s + (−4.86 − 4.08i)7-s + (−2.73 − 0.732i)8-s + (2.81 + 1.02i)9-s + (−6.57 + 11.3i)10-s + (−3.84 + 2.21i)11-s + (−0.601 − 3.41i)12-s + (−1.45 + 3.12i)13-s + (−8.67 + 2.32i)14-s + (−16.0 − 1.40i)15-s + (−3.06 + 2.57i)16-s + (−20.3 + 9.51i)17-s + ⋯ |
| L(s) = 1 | + (0.405 − 0.579i)2-s + (0.568 + 0.100i)3-s + (−0.171 − 0.469i)4-s + (−1.85 + 0.162i)5-s + (0.288 − 0.288i)6-s + (−0.695 − 0.583i)7-s + (−0.341 − 0.0915i)8-s + (0.313 + 0.114i)9-s + (−0.657 + 1.13i)10-s + (−0.349 + 0.201i)11-s + (−0.0501 − 0.284i)12-s + (−0.112 + 0.240i)13-s + (−0.619 + 0.166i)14-s + (−1.06 − 0.0935i)15-s + (−0.191 + 0.160i)16-s + (−1.19 + 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0557262 + 0.367412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0557262 + 0.367412i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.811 + 1.15i)T \) |
| 3 | \( 1 + (-1.70 - 0.300i)T \) |
| 37 | \( 1 + (-18.7 + 31.9i)T \) |
| good | 5 | \( 1 + (9.26 - 0.810i)T + (24.6 - 4.34i)T^{2} \) |
| 7 | \( 1 + (4.86 + 4.08i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (3.84 - 2.21i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.45 - 3.12i)T + (-108. - 129. i)T^{2} \) |
| 17 | \( 1 + (20.3 - 9.51i)T + (185. - 221. i)T^{2} \) |
| 19 | \( 1 + (2.92 + 4.17i)T + (-123. + 339. i)T^{2} \) |
| 23 | \( 1 + (5.81 + 21.6i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-6.47 + 24.1i)T + (-728. - 420.5i)T^{2} \) |
| 31 | \( 1 + (29.7 + 29.7i)T + 961iT^{2} \) |
| 41 | \( 1 + (-20.0 - 55.0i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-14.6 + 14.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (32.2 - 55.7i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (23.4 - 19.7i)T + (487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-7.11 + 81.3i)T + (-3.42e3 - 604. i)T^{2} \) |
| 61 | \( 1 + (-47.4 - 22.1i)T + (2.39e3 + 2.85e3i)T^{2} \) |
| 67 | \( 1 + (50.2 - 59.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (2.87 - 16.3i)T + (-4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 - 27.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-88.7 + 7.76i)T + (6.14e3 - 1.08e3i)T^{2} \) |
| 83 | \( 1 + (55.4 + 20.1i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (126. + 11.0i)T + (7.80e3 + 1.37e3i)T^{2} \) |
| 97 | \( 1 + (-15.4 - 57.5i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.40335311575861789515136416064, −10.83630751721592490300872758619, −9.674882169261859858095040994607, −8.485697530749345024412960912257, −7.56255014612169709449984191441, −6.52909414981526769796934003850, −4.40801257311949984679343967739, −3.95505636580547815392234144615, −2.67435196518503118293167185263, −0.16003013697424116476980802273,
2.98967431376029969955706315883, 3.91749656388843419654573184966, 5.13950460967788398628415511636, 6.74228126093281249720459730367, 7.56162923557387773363027110932, 8.459929764699348635349567244288, 9.202834008546406969719936863931, 10.85853348733515151413176943068, 11.89859200436926133269591366936, 12.59454368170162732882153845665
