L(s) = 1 | + (−0.817 − 1.41i)2-s + (1.40 − 1.01i)3-s + (−0.335 + 0.580i)4-s + (0.5 − 0.866i)5-s + (−2.58 − 1.15i)6-s + (−1.06 − 1.84i)7-s − 2.17·8-s + (0.940 − 2.84i)9-s − 1.63·10-s + (0.0263 + 0.0455i)11-s + (0.118 + 1.15i)12-s + (0.5 − 0.866i)13-s + (−1.74 + 3.01i)14-s + (−0.177 − 1.72i)15-s + (2.44 + 4.23i)16-s − 2.48·17-s + ⋯ |
L(s) = 1 | + (−0.577 − 1.00i)2-s + (0.810 − 0.585i)3-s + (−0.167 + 0.290i)4-s + (0.223 − 0.387i)5-s + (−1.05 − 0.472i)6-s + (−0.402 − 0.697i)7-s − 0.768·8-s + (0.313 − 0.949i)9-s − 0.516·10-s + (0.00793 + 0.0137i)11-s + (0.0342 + 0.333i)12-s + (0.138 − 0.240i)13-s + (−0.465 + 0.805i)14-s + (−0.0457 − 0.444i)15-s + (0.611 + 1.05i)16-s − 0.601·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0909839 + 1.25960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0909839 + 1.25960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.40 + 1.01i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.817 + 1.41i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1.06 + 1.84i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0263 - 0.0455i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + (2.46 - 4.27i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.24 - 2.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.08 + 7.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + (-2.73 + 4.74i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.73 - 8.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.88 + 8.45i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.89 - 5.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 + (-7.80 - 13.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.244 + 0.423i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 4.64T + 89T^{2} \) |
| 97 | \( 1 + (3.67 + 6.35i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.996795322975517636340017169636, −9.560247308180284459206293117470, −8.696447049286355753560512019980, −7.82833509529398232957365175246, −6.78852097814854108497509489667, −5.79657792813570335557505567564, −4.08005277085453483545678933728, −3.07655464688617898193993376958, −1.95659681196968179459465901619, −0.77459067520787707870265516409,
2.41285575460481785083753865394, 3.32844117715815408255508347222, 4.75577102133660786306433731446, 6.00168904269550221293165837414, 6.74596477818781484733939579695, 7.77373216252023511951632034970, 8.549786191639792693865859970052, 9.200778059427653314500479300205, 9.887689180120064025935094765458, 10.86564020521535431111525887786