L(s) = 1 | + (−0.186 + 0.0606i)2-s + (−0.865 − 2.66i)3-s + (−1.58 + 1.15i)4-s + (2.93 − 2.13i)5-s + (0.322 + 0.444i)6-s + (−0.222 − 0.0723i)7-s + (0.456 − 0.628i)8-s + (−3.91 + 2.84i)9-s + (−0.417 + 0.575i)10-s + 5.67i·11-s + (4.44 + 3.22i)12-s + 2.04·13-s + 0.0459·14-s + (−8.20 − 5.96i)15-s + (1.16 − 3.58i)16-s + (1.75 + 2.41i)17-s + ⋯ |
L(s) = 1 | + (−0.131 + 0.0428i)2-s + (−0.499 − 1.53i)3-s + (−0.793 + 0.576i)4-s + (1.31 − 0.952i)5-s + (0.131 + 0.181i)6-s + (−0.0841 − 0.0273i)7-s + (0.161 − 0.222i)8-s + (−1.30 + 0.947i)9-s + (−0.132 + 0.181i)10-s + 1.71i·11-s + (1.28 + 0.931i)12-s + 0.565·13-s + 0.0122·14-s + (−2.11 − 1.53i)15-s + (0.291 − 0.896i)16-s + (0.425 + 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.616255 - 0.407747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.616255 - 0.407747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 + (7.54 + 2.01i)T \) |
good | 2 | \( 1 + (0.186 - 0.0606i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.865 + 2.66i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.93 + 2.13i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.0723i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 - 5.67iT - 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + (-1.75 - 2.41i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.260 + 0.801i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.06 + 2.84i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 - 3.91iT - 29T^{2} \) |
| 31 | \( 1 + (3.42 + 1.11i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.13 + 0.368i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.18 - 3.65i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.62 - 7.74i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 + (1.35 - 1.86i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.75 + 1.54i)T + (47.7 - 34.6i)T^{2} \) |
| 67 | \( 1 + (1.99 + 2.74i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (1.12 - 1.55i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.14 + 3.01i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.05 + 8.33i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.02 + 9.32i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.67 + 2.16i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.58 + 11.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.39205846864938031524952053712, −13.21426826370634321047100860981, −12.86319770983173924433676212848, −12.10905680943220180176899470065, −10.02987113400393642598632011248, −8.867744056598731147559522924949, −7.63788518839913418535350083823, −6.28601828688923696022051831601, −4.90488961792261668237958826854, −1.68183952835725434680805745947,
3.55115321486051542253146341599, 5.43166616976039355140778770504, 6.02859172794265792170980537889, 8.820453785011287534860333014533, 9.761736100951160921822227907588, 10.48984340512639276876068221685, 11.25623578539732775881486783918, 13.64947038541836528637633571709, 14.09363171315168027701664179721, 15.23973070537076784224102490824