| L(s) = 1 | + (−0.207 + 1.97i)2-s + (1.86 + 2.07i)3-s + (−1.89 − 0.402i)4-s + (−0.0245 − 0.0109i)5-s + (−4.47 + 3.25i)6-s + (−0.0378 + 0.116i)8-s + (−0.497 + 4.73i)9-s + (0.0267 − 0.0462i)10-s + (3.30 + 0.316i)11-s + (−2.70 − 4.67i)12-s + (3.94 + 2.86i)13-s + (−0.0231 − 0.0713i)15-s + (−3.76 − 1.67i)16-s + (−0.175 − 1.66i)17-s + (−9.24 − 1.96i)18-s + (−1.34 + 0.285i)19-s + ⋯ |
| L(s) = 1 | + (−0.146 + 1.39i)2-s + (1.07 + 1.19i)3-s + (−0.947 − 0.201i)4-s + (−0.0109 − 0.00489i)5-s + (−1.82 + 1.32i)6-s + (−0.0133 + 0.0411i)8-s + (−0.165 + 1.57i)9-s + (0.00844 − 0.0146i)10-s + (0.995 + 0.0955i)11-s + (−0.779 − 1.35i)12-s + (1.09 + 0.795i)13-s + (−0.00598 − 0.0184i)15-s + (−0.940 − 0.418i)16-s + (−0.0425 − 0.404i)17-s + (−2.17 − 0.463i)18-s + (−0.308 + 0.0655i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0532588 - 1.94216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0532588 - 1.94216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.30 - 0.316i)T \) |
| good | 2 | \( 1 + (0.207 - 1.97i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-1.86 - 2.07i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (0.0245 + 0.0109i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-3.94 - 2.86i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.175 + 1.66i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.34 - 0.285i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (4.03 + 6.98i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.97 + 6.08i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.66 + 1.63i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.348 + 0.387i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (3.27 - 10.0i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + (-8.76 + 1.86i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (3.61 - 1.61i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (9.51 + 2.02i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.73 - 3.44i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (1.40 - 2.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.65 - 1.20i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-10.2 - 2.17i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.612 - 5.82i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (2.10 - 1.53i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (0.608 + 1.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.74 + 1.99i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.07656787650464976388161235244, −9.905261374108609007708383752830, −9.259376895432894958489948932065, −8.490184198735294852072295230417, −8.007980625336953906041943083995, −6.65437358573047089614382135841, −5.99958392006134334799789691660, −4.46264306698850227789700139280, −4.04362737563123833201698644963, −2.43911514609466722866778269718,
1.20812697135146582183377121336, 1.95806555349226173441794063854, 3.30398882520937039506856271935, 3.80405332708683366327075833905, 5.88950316322897913038097365690, 6.91912617692106141057944445545, 7.905043586058683948857406113253, 8.816814893281009418792645795898, 9.360402153527553330293793447395, 10.51491843522518181793483025244
