| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.69 + 0.355i)3-s + (−0.939 − 0.342i)4-s + (−0.882 − 2.42i)5-s + (0.0553 + 1.73i)6-s + (−1.58 − 2.74i)7-s + (−0.5 + 0.866i)8-s + (2.74 − 1.20i)9-s + (−2.54 + 0.448i)10-s + (−2.16 − 1.25i)11-s + (1.71 + 0.246i)12-s + (2.71 + 3.24i)13-s + (−2.97 + 1.08i)14-s + (2.35 + 3.79i)15-s + (0.766 + 0.642i)16-s + (1.32 + 0.233i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.978 + 0.205i)3-s + (−0.469 − 0.171i)4-s + (−0.394 − 1.08i)5-s + (0.0225 + 0.706i)6-s + (−0.598 − 1.03i)7-s + (−0.176 + 0.306i)8-s + (0.915 − 0.401i)9-s + (−0.803 + 0.141i)10-s + (−0.653 − 0.377i)11-s + (0.494 + 0.0710i)12-s + (0.754 + 0.898i)13-s + (−0.795 + 0.289i)14-s + (0.608 + 0.980i)15-s + (0.191 + 0.160i)16-s + (0.320 + 0.0565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.241355 - 0.593517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241355 - 0.593517i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.69 - 0.355i)T \) |
| 19 | \( 1 + (-3.14 + 3.01i)T \) |
| good | 5 | \( 1 + (0.882 + 2.42i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.58 + 2.74i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.16 + 1.25i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.71 - 3.24i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 0.233i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.30 - 3.58i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.32 + 7.49i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-6.89 + 3.97i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.10iT - 37T^{2} \) |
| 41 | \( 1 + (-4.95 - 4.16i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (11.7 - 4.27i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.16 + 1.08i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.46 + 1.26i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.54 - 8.75i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.133 + 0.0485i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.48 - 0.791i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-8.59 + 3.12i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.67 + 1.40i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.41 + 7.64i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.3 + 7.11i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (12.7 - 10.6i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.538 - 0.0949i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.21090281648750660169401517775, −11.96760656956521964946211480237, −11.30921141009788960968144741533, −10.18120381341094999144470627606, −9.276209455572470696759049752050, −7.75476612057040347973778324229, −6.18126920669865048978847474985, −4.80092539058265098982760743232, −3.83073855748623438737217431605, −0.811678746607932647894916086709,
3.20806724500156506145965291447, 5.21352238733988797866486616341, 6.16852533958799680590282538300, 7.12266382131761641074341282271, 8.280536776008512771241218123211, 9.972874864985412240576743943625, 10.82022482476527321203791452373, 12.11884503856924290770218776446, 12.77050695885013944320113102445, 14.07326300619033799573469171471
