L(s) = 1 | + (0.190 − 0.587i)2-s + (1.30 + 0.951i)4-s + (0.809 + 2.48i)5-s + (−2.42 − 1.76i)7-s + (1.80 − 1.31i)8-s + 1.61·10-s + (−1.69 − 2.85i)11-s + (0.545 − 1.67i)13-s + (−1.5 + 1.08i)14-s + (0.572 + 1.76i)16-s + (−0.5 − 1.53i)17-s + (−4.73 + 3.44i)19-s + (−1.30 + 4.02i)20-s + (−2 + 0.449i)22-s − 3.47·23-s + ⋯ |
L(s) = 1 | + (0.135 − 0.415i)2-s + (0.654 + 0.475i)4-s + (0.361 + 1.11i)5-s + (−0.917 − 0.666i)7-s + (0.639 − 0.464i)8-s + 0.511·10-s + (−0.509 − 0.860i)11-s + (0.151 − 0.465i)13-s + (−0.400 + 0.291i)14-s + (0.143 + 0.440i)16-s + (−0.121 − 0.373i)17-s + (−1.08 + 0.789i)19-s + (−0.292 + 0.900i)20-s + (−0.426 + 0.0957i)22-s − 0.723·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18014 - 0.0280619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18014 - 0.0280619i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.69 + 2.85i)T \) |
good | 2 | \( 1 + (-0.190 + 0.587i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.809 - 2.48i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (2.42 + 1.76i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.545 + 1.67i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.881 + 2.71i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (9.66 - 7.02i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + (-1.30 + 0.951i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.97 + 9.14i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.35 - 6.06i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.42 - 7.46i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + (-1.71 - 5.29i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.92 + 9.00i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.218 + 0.673i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 0.527T + 89T^{2} \) |
| 97 | \( 1 + (4.33 - 13.3i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.64976981721785308814852644527, −12.94933566608028165844851750419, −11.68294092597360301387349458332, −10.51361359276888462709231394160, −10.21656361328303341069277793850, −8.215873627560383150253341548128, −6.95837837767768320332728360660, −6.13731191355275439606919306151, −3.70461679290982177190831233442, −2.67353381586304762334914147412,
2.17771360057825058654361214910, 4.69391520472387922687432997586, 5.86792800328731985688302816538, 6.87911670244084392931576428502, 8.465345796978454265012026473221, 9.535510910893011390947133214239, 10.58136597216823745583855793868, 12.08559870458602472863358782503, 12.81953525050876356491018718170, 13.88441208634309124846948143435