| L(s) = 1 | + (0.678 + 1.24i)2-s + (−1.70 − 0.301i)3-s + (−1.08 + 1.68i)4-s + (−0.000967 − 0.00110i)5-s + (−0.782 − 2.32i)6-s + (−1.80 − 0.238i)7-s + (−2.82 − 0.198i)8-s + (2.81 + 1.02i)9-s + (0.000713 − 0.00194i)10-s + (0.208 + 0.421i)11-s + (2.34 − 2.54i)12-s + (−1.52 − 4.50i)13-s + (−0.930 − 2.40i)14-s + (0.00131 + 0.00217i)15-s + (−1.66 − 3.63i)16-s + (−4.32 − 4.32i)17-s + ⋯ |
| L(s) = 1 | + (0.479 + 0.877i)2-s + (−0.984 − 0.173i)3-s + (−0.540 + 0.841i)4-s + (−0.000432 − 0.000493i)5-s + (−0.319 − 0.947i)6-s + (−0.683 − 0.0899i)7-s + (−0.997 − 0.0703i)8-s + (0.939 + 0.342i)9-s + (0.000225 − 0.000616i)10-s + (0.0627 + 0.127i)11-s + (0.678 − 0.734i)12-s + (−0.424 − 1.24i)13-s + (−0.248 − 0.642i)14-s + (0.000340 + 0.000561i)15-s + (−0.416 − 0.909i)16-s + (−1.04 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.333651 - 0.250179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333651 - 0.250179i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.678 - 1.24i)T \) |
| 3 | \( 1 + (1.70 + 0.301i)T \) |
| good | 5 | \( 1 + (0.000967 + 0.00110i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (1.80 + 0.238i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-0.208 - 0.421i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.52 + 4.50i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.32 + 4.32i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.25 - 6.28i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.497 + 3.77i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-8.13 + 0.533i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-3.92 + 2.26i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.838 + 4.21i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.920 + 6.99i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (7.91 - 3.90i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (6.62 - 1.77i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.66 - 6.98i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (6.01 + 6.85i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.683 - 10.4i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-5.08 - 2.50i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (2.87 + 6.93i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (4.91 - 11.8i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.125 + 0.469i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (10.2 + 8.99i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-3.50 + 1.45i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-1.67 - 0.966i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.40633595565492336757676898822, −9.841068466229504283822324925350, −8.496970980426502412637107366973, −7.64593735311001034847197563314, −6.62427282584225783060216485998, −6.15261067155942817281878771551, −5.06110687605061663310227204878, −4.26772714934772418332176834633, −2.80974285950358293244189207283, −0.22517344607939372962281259801,
1.64854391306657935787923660699, 3.19946934303676040530122987717, 4.43495666584177949161797676627, 5.03093541740136577722565356567, 6.52102471290557587603649285632, 6.61605458299042114308115627979, 8.634273201910054171254962854410, 9.520375858078921963190927693466, 10.14897984617978783535125873608, 11.20465194218228238511541271238
