L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.331 + 0.107i)3-s + (−0.309 − 0.951i)4-s + (−1.25 − 1.84i)5-s + (−0.107 + 0.331i)6-s − i·7-s + (−0.951 − 0.309i)8-s + (−2.32 + 1.69i)9-s + (−2.23 − 0.0683i)10-s + (−2.18 − 1.58i)11-s + (0.204 + 0.281i)12-s + (−2.81 − 3.87i)13-s + (−0.809 − 0.587i)14-s + (0.616 + 0.477i)15-s + (−0.809 + 0.587i)16-s + (5.80 + 1.88i)17-s + ⋯ |
L(s) = 1 | + (0.415 − 0.572i)2-s + (−0.191 + 0.0621i)3-s + (−0.154 − 0.475i)4-s + (−0.562 − 0.826i)5-s + (−0.0439 + 0.135i)6-s − 0.377i·7-s + (−0.336 − 0.109i)8-s + (−0.776 + 0.563i)9-s + (−0.706 − 0.0216i)10-s + (−0.658 − 0.478i)11-s + (0.0591 + 0.0813i)12-s + (−0.781 − 1.07i)13-s + (−0.216 − 0.157i)14-s + (0.159 + 0.123i)15-s + (−0.202 + 0.146i)16-s + (1.40 + 0.457i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.210387 - 0.937491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.210387 - 0.937491i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (1.25 + 1.84i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.331 - 0.107i)T + (2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (2.18 + 1.58i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.81 + 3.87i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.80 - 1.88i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.15 + 3.55i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.879 - 1.21i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.88 + 8.87i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.33 - 4.12i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.24 + 3.08i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-9.11 + 6.62i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.26iT - 43T^{2} \) |
| 47 | \( 1 + (-10.5 + 3.42i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.496 + 0.161i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.82 + 2.05i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.73 + 1.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-13.2 - 4.32i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.84 + 8.75i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.88 - 5.34i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.21 - 9.88i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.89 - 2.56i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (1.96 + 1.42i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.32 + 1.08i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.13532167976840428567117379547, −10.42129998483292929599016467266, −9.395002371497594382558263590497, −8.151169066277423777746915136123, −7.58881789940893121098226579271, −5.62773593136791638673911492593, −5.23465970531273914779749794043, −3.90291411890134503860257316053, −2.67915729243788550445732864721, −0.58221858314141814870849736676,
2.65708018560552357632620116662, 3.76929682212897332052742671633, 5.15084845245999202563450735970, 6.07925008685909663534592661313, 7.17367790181059761872999630954, 7.77756387576551526672574140200, 9.022532477408984044559373417098, 10.01947036892220071686843989163, 11.18805512013304112256719389609, 12.10321042144166667808053982081