| L(s) = 1 | + (−1.12 + 0.694i)2-s + (0.934 + 3.28i)3-s + (−0.116 + 0.234i)4-s + (1.26 − 0.491i)5-s + (−3.32 − 3.03i)6-s + (0.179 − 1.94i)7-s + (−0.275 − 2.96i)8-s + (−7.36 + 4.56i)9-s + (−1.08 + 1.43i)10-s + (3.98 − 2.46i)11-s + (−0.878 − 0.164i)12-s + (−0.157 + 1.70i)13-s + (1.14 + 2.30i)14-s + (2.80 + 3.70i)15-s + (2.05 + 2.71i)16-s + (0.580 − 0.528i)17-s + ⋯ |
| L(s) = 1 | + (−0.792 + 0.490i)2-s + (0.539 + 1.89i)3-s + (−0.0583 + 0.117i)4-s + (0.567 − 0.219i)5-s + (−1.35 − 1.23i)6-s + (0.0679 − 0.733i)7-s + (−0.0972 − 1.04i)8-s + (−2.45 + 1.52i)9-s + (−0.341 + 0.452i)10-s + (1.20 − 0.744i)11-s + (−0.253 − 0.0474i)12-s + (−0.0438 + 0.472i)13-s + (0.306 + 0.614i)14-s + (0.723 + 0.957i)15-s + (0.513 + 0.679i)16-s + (0.140 − 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.365701 + 0.742020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.365701 + 0.742020i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (-1.02 + 10.0i)T \) |
| good | 2 | \( 1 + (1.12 - 0.694i)T + (0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-0.934 - 3.28i)T + (-2.55 + 1.57i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 0.491i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.179 + 1.94i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (-3.98 + 2.46i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.157 - 1.70i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (-0.580 + 0.528i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.370 - 1.30i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (3.48 + 2.15i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-8.85 + 3.42i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 1.55i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 0.383i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (4.55 + 1.76i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (5.63 - 1.05i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 - 0.973T + 47T^{2} \) |
| 53 | \( 1 + (2.36 - 8.29i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (1.26 + 13.6i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-2.66 + 2.43i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (0.433 + 4.67i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (2.42 + 0.937i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (7.08 + 2.74i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-2.00 + 0.775i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-0.500 + 5.39i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-1.58 - 3.17i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (4.80 + 4.37i)T + (8.95 + 96.5i)T^{2} \) |
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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
−14.20525024934894799200633315347, −13.75020587969569050932016653034, −11.74781283698427858562662365936, −10.42036932515678518139969278618, −9.706963562213174976117609314410, −8.933947896281947039486197413571, −8.104236030280828405362187825657, −6.25625291256694123819885984441, −4.49858617674242054642637732004, −3.53151462382935858123664522666,
1.48327612824794099833565749642, 2.56612686197302527753129552798, 5.78119717333806570274930693409, 6.78503715491892990305883313667, 8.158790603836630033144217846520, 8.931492759830087570973615954263, 9.989368892848561786174994821421, 11.70660557355577713966068379493, 12.16999765973546647119328431417, 13.45021419309614307387320304190
