| L(s) = 1 | − 5-s − 5.89i·11-s + 4.65i·13-s − 2.10·17-s + 5.25i·19-s + 7.83i·23-s + 25-s + 0.273i·29-s − 4.93i·31-s + 5.83·37-s − 2.26·41-s + 8.63·43-s − 4.98·47-s − 5.62i·53-s + 5.89i·55-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.77i·11-s + 1.29i·13-s − 0.509·17-s + 1.20i·19-s + 1.63i·23-s + 0.200·25-s + 0.0508i·29-s − 0.885i·31-s + 0.959·37-s − 0.353·41-s + 1.31·43-s − 0.727·47-s − 0.772i·53-s + 0.795i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2256197863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2256197863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5.89iT - 11T^{2} \) |
| 13 | \( 1 - 4.65iT - 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 - 7.83iT - 23T^{2} \) |
| 29 | \( 1 - 0.273iT - 29T^{2} \) |
| 31 | \( 1 + 4.93iT - 31T^{2} \) |
| 37 | \( 1 - 5.83T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 8.63T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 5.62iT - 53T^{2} \) |
| 59 | \( 1 + 4.67T + 59T^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 14.3iT - 71T^{2} \) |
| 73 | \( 1 - 0.741iT - 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 - 6.81T + 89T^{2} \) |
| 97 | \( 1 + 4.90iT - 97T^{2} \) |
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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
−7.930213063244031624659265652914, −7.61309931755229127336498692979, −6.54139319965024319852774112749, −6.09557505580700330566044245030, −5.41773057811038399299189155400, −4.45847386454445489983595771257, −3.74044901519537018754384783789, −3.25325766865318926882387060284, −2.11339758511298202383221160718, −1.18112562879885257544446108760,
0.05610106808497856506895067212, 1.19801587135602101741384484147, 2.48327374027496102317068482290, 2.84097294807955783441657169814, 4.13553297762374747562062208105, 4.56773289208786855851101796605, 5.19371235365489653493682188212, 6.15799770706942356420008282738, 6.93494167862532266429634995139, 7.33864362406410046104890539316
