L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s − 2i·13-s − 14-s + 16-s − 2·19-s + 2·26-s − i·28-s + 6·29-s + 8·31-s + i·32-s − 4i·37-s − 2i·38-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.458·19-s + 0.392·26-s − 0.188i·28-s + 1.11·29-s + 1.43·31-s + 0.176i·32-s − 0.657i·37-s − 0.324i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.648116311\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648116311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.538725681242857869666908292632, −8.200121323913554069772071555711, −7.28002282381830107871109533855, −6.52300765544816658970992038916, −5.88210687213051791097291490817, −5.06083094473874519121372982960, −4.33758145465434388558654496862, −3.29071773038933164007453995990, −2.31612230040998051861296662072, −0.830246319784123610551317756921,
0.73863750808720447045536713662, 1.89248978017999300971331797570, 2.84891059370008941722530675145, 3.79068177135856846222756450670, 4.54944079256386358730642439459, 5.26336648189400356114605460107, 6.44244083100468576861231443861, 6.90582404766489243037836502391, 8.122393540218572032822276816006, 8.482748299997655803667908699491