| L(s) = 1 | + (−0.755 + 0.654i)2-s + (1.53 − 2.39i)3-s + (0.142 − 0.989i)4-s + (2.02 + 0.949i)5-s + (0.405 + 2.81i)6-s + (0.910 − 3.10i)7-s + (0.540 + 0.841i)8-s + (−2.12 − 4.64i)9-s + (−2.15 + 0.608i)10-s + (−3.94 + 4.55i)11-s + (−2.15 − 1.86i)12-s + (0.340 + 1.16i)13-s + (1.34 + 2.93i)14-s + (5.38 − 3.38i)15-s + (−0.959 − 0.281i)16-s + (−1.29 + 0.185i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 + 0.463i)2-s + (0.888 − 1.38i)3-s + (0.0711 − 0.494i)4-s + (0.905 + 0.424i)5-s + (0.165 + 1.15i)6-s + (0.344 − 1.17i)7-s + (0.191 + 0.297i)8-s + (−0.706 − 1.54i)9-s + (−0.680 + 0.192i)10-s + (−1.18 + 1.37i)11-s + (−0.621 − 0.538i)12-s + (0.0945 + 0.321i)13-s + (0.358 + 0.785i)14-s + (1.39 − 0.874i)15-s + (−0.239 − 0.0704i)16-s + (−0.313 + 0.0450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26347 - 0.538237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26347 - 0.538237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.755 - 0.654i)T \) |
| 5 | \( 1 + (-2.02 - 0.949i)T \) |
| 23 | \( 1 + (-2.38 + 4.15i)T \) |
| good | 3 | \( 1 + (-1.53 + 2.39i)T + (-1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-0.910 + 3.10i)T + (-5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.94 - 4.55i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.340 - 1.16i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.29 - 0.185i)T + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.209 + 1.45i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.985 - 6.85i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.03 + 3.23i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (10.2 - 4.68i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (0.408 - 0.894i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.896 + 1.39i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.01iT - 47T^{2} \) |
| 53 | \( 1 + (-2.48 + 8.44i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (5.17 - 1.51i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (11.8 - 7.58i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.16 + 3.61i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.06 - 3.54i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-6.25 - 0.899i)T + (70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (3.31 - 0.973i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.802 + 0.366i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.01 - 2.58i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.06 + 0.944i)T + (63.5 + 73.3i)T^{2} \) |
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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
−12.43820279875715546238594451372, −10.79614961920946346056956388047, −10.08272678119157152064090514428, −8.932627126392005125244465912038, −7.88380851534702707003829887661, −7.09732217055660332730098240698, −6.62172722885023599348818709787, −4.88151353809193979956388438966, −2.66073427167100434601897615320, −1.52343703851190643093065799114,
2.35146484101113320849521246494, 3.29248727281009153972274839854, 4.97502339225392418636755813802, 5.78204010541238345349531047765, 8.108698584543415550097846705791, 8.681015317716140408816842207965, 9.341044843418925757071520778274, 10.28463489195632447027444627036, 10.94266690078158666701786976922, 12.19976785068846387915240710524
