| L(s) = 1 | + (1.26 − 0.642i)2-s + (−0.312 + 1.97i)3-s + (1.17 − 1.61i)4-s + (−1.25 + 4.83i)5-s + (0.874 + 2.69i)6-s + (1.37 + 8.65i)7-s + (0.442 − 2.79i)8-s + (4.75 + 1.54i)9-s + (1.52 + 6.90i)10-s + (6.66 − 8.75i)11-s + (2.82 + 2.82i)12-s + (−5.96 − 11.6i)13-s + (7.28 + 10.0i)14-s + (−9.17 − 3.99i)15-s + (−1.23 − 3.80i)16-s + (−4.90 + 9.62i)17-s + ⋯ |
| L(s) = 1 | + (0.630 − 0.321i)2-s + (−0.104 + 0.658i)3-s + (0.293 − 0.404i)4-s + (−0.251 + 0.967i)5-s + (0.145 + 0.448i)6-s + (0.195 + 1.23i)7-s + (0.0553 − 0.349i)8-s + (0.528 + 0.171i)9-s + (0.152 + 0.690i)10-s + (0.606 − 0.795i)11-s + (0.235 + 0.235i)12-s + (−0.458 − 0.899i)13-s + (0.520 + 0.716i)14-s + (−0.611 − 0.266i)15-s + (−0.0772 − 0.237i)16-s + (−0.288 + 0.566i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.69985 + 0.712788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69985 + 0.712788i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.642i)T \) |
| 5 | \( 1 + (1.25 - 4.83i)T \) |
| 11 | \( 1 + (-6.66 + 8.75i)T \) |
| good | 3 | \( 1 + (0.312 - 1.97i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.37 - 8.65i)T + (-46.6 + 15.1i)T^{2} \) |
| 13 | \( 1 + (5.96 + 11.6i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (4.90 - 9.62i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-1.08 - 1.49i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-10.9 + 10.9i)T - 529iT^{2} \) |
| 29 | \( 1 + (-12.7 + 17.5i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-14.1 + 43.6i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (4.47 + 28.2i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (59.4 - 43.1i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-0.193 + 0.193i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-45.2 - 7.16i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-18.0 - 35.4i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-62.2 + 85.7i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (18.7 + 57.8i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-47.1 - 47.1i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (6.78 + 20.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (56.4 - 8.93i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (33.4 + 10.8i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-95.8 - 48.8i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (153. - 78.4i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−13.58046027768028331535761055195, −12.38906344411623305324100946506, −11.42782548434718973117866447771, −10.60394737269649445441355702411, −9.580316557605276336457827860700, −8.116522332659042244013695673697, −6.50876602852254424328641819477, −5.39819594224469862805481331580, −3.93405325834908591268343695389, −2.58009789551101647048228533321,
1.40230738585879170738306851249, 4.05997845984385095130024350385, 4.91336805046840606868048929417, 6.92633998250359555097670871036, 7.23849024303769818516824857938, 8.789035388132606718014045320866, 10.11577322124024791985170155375, 11.74089961840004863242384786822, 12.31153302181225558159752022959, 13.38435764098737583551056160025
