L(s) = 1 | + (1.96 − 1.96i)2-s − 5.71i·4-s + (1.65 + 1.50i)5-s + (0.707 + 0.707i)7-s + (−7.29 − 7.29i)8-s + (6.20 − 0.280i)10-s + 0.248i·11-s + (−3.37 + 3.37i)13-s + 2.77·14-s − 17.2·16-s + (1.75 − 1.75i)17-s + 3.91i·19-s + (8.62 − 9.43i)20-s + (0.487 + 0.487i)22-s + (−2.22 − 2.22i)23-s + ⋯ |
L(s) = 1 | + (1.38 − 1.38i)2-s − 2.85i·4-s + (0.738 + 0.674i)5-s + (0.267 + 0.267i)7-s + (−2.58 − 2.58i)8-s + (1.96 − 0.0886i)10-s + 0.0749i·11-s + (−0.935 + 0.935i)13-s + 0.742·14-s − 4.31·16-s + (0.426 − 0.426i)17-s + 0.898i·19-s + (1.92 − 2.11i)20-s + (0.104 + 0.104i)22-s + (−0.463 − 0.463i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55859 - 2.24381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55859 - 2.24381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.65 - 1.50i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.96 + 1.96i)T - 2iT^{2} \) |
| 11 | \( 1 - 0.248iT - 11T^{2} \) |
| 13 | \( 1 + (3.37 - 3.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.75 + 1.75i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.91iT - 19T^{2} \) |
| 23 | \( 1 + (2.22 + 2.22i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.42iT - 41T^{2} \) |
| 43 | \( 1 + (0.716 - 0.716i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.34 + 3.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.25 - 4.25i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 + (5.98 + 5.98i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.31iT - 71T^{2} \) |
| 73 | \( 1 + (10.3 - 10.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.2iT - 79T^{2} \) |
| 83 | \( 1 + (10.1 + 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 + (-1.07 - 1.07i)T + 97iT^{2} \) |
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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
−11.67959066492389932227328617456, −10.54743699881453884989981485032, −10.00545217185730012760765095547, −9.127455872387642763229634114731, −7.06317463004377795755351039232, −6.00310060176396888746648703283, −5.15922647110697978720616795637, −4.02026810216807406704887642899, −2.72323225185120017843992028112, −1.82537741977140221309124362915,
2.72598295437535964102393127303, 4.19517220664973877874455251253, 5.16960198500875387800868699064, 5.78052081113694880124021110923, 6.93779391102343111599622167028, 7.86287776195805363553294965222, 8.677737967061111437757700365139, 9.918771680474788042537607478185, 11.46410200934367839004381732344, 12.49316784241039269496246096887