L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.49 − 1.66i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−2.23 + 0.122i)10-s − 0.520·11-s + (0.707 − 0.707i)12-s + (2.39 + 2.39i)13-s + (−2.23 + 0.122i)15-s − 1.00·16-s + (−0.110 + 0.110i)17-s + (0.707 − 0.707i)18-s + 6.73·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.667 − 0.744i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.706 + 0.0386i)10-s − 0.156·11-s + (0.204 − 0.204i)12-s + (0.663 + 0.663i)13-s + (−0.576 + 0.0315i)15-s − 0.250·16-s + (−0.0267 + 0.0267i)17-s + (0.166 − 0.166i)18-s + 1.54·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.344605879\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344605879\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.49 + 1.66i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.520T + 11T^{2} \) |
| 13 | \( 1 + (-2.39 - 2.39i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.110 - 0.110i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + (-0.802 + 0.802i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.20iT - 29T^{2} \) |
| 31 | \( 1 - 7.18iT - 31T^{2} \) |
| 37 | \( 1 + (-4.41 - 4.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.23iT - 41T^{2} \) |
| 43 | \( 1 + (-6.27 + 6.27i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.57 + 7.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.550 + 0.550i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.93T + 59T^{2} \) |
| 61 | \( 1 - 15.4iT - 61T^{2} \) |
| 67 | \( 1 + (10.4 + 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 + (-3.22 - 3.22i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.29iT - 79T^{2} \) |
| 83 | \( 1 + (3.43 + 3.43i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + (-9.40 + 9.40i)T - 97iT^{2} \) |
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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
−9.227493025379984313318968702309, −8.791375601720376044743843741833, −7.81782877740661427559745128246, −6.98737369172027260602232437211, −6.03286587829133122580757393863, −5.23394164653959110980004926051, −4.27352586600873827470183411937, −2.98048873326306024690373165462, −1.76787130691627927487316840522, −0.899025128278616773380753080078,
1.04109045073573287789850090334, 2.58377513223097753871348401518, 3.63532242155744052867296199761, 4.94314883650609559124876878215, 5.84213087499922907811070005687, 6.19810416456475013561886774745, 7.38691820930490992840648735230, 7.84667680771227923452414623968, 9.220952040674975629393224403923, 9.500932756695375498729409193733