L(s) = 1 | + (0.696 + 1.87i)2-s + (−3.02 + 2.61i)4-s + 5.46·7-s + (−7.00 − 3.86i)8-s − 11.0i·11-s + 10.1i·13-s + (3.80 + 10.2i)14-s + (2.35 − 15.8i)16-s − 24.4i·17-s − 23.7i·19-s + (20.6 − 7.69i)22-s − 37.2·23-s + (−18.9 + 7.05i)26-s + (−16.5 + 14.2i)28-s − 25.7·29-s + ⋯ |
L(s) = 1 | + (0.348 + 0.937i)2-s + (−0.757 + 0.652i)4-s + 0.781·7-s + (−0.875 − 0.482i)8-s − 1.00i·11-s + 0.778i·13-s + (0.272 + 0.732i)14-s + (0.147 − 0.989i)16-s − 1.43i·17-s − 1.25i·19-s + (0.940 − 0.349i)22-s − 1.61·23-s + (−0.730 + 0.271i)26-s + (−0.591 + 0.510i)28-s − 0.888·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.552887411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.552887411\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.696 - 1.87i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.46T + 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 - 10.1iT - 169T^{2} \) |
| 17 | \( 1 + 24.4iT - 289T^{2} \) |
| 19 | \( 1 + 23.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2T + 529T^{2} \) |
| 29 | \( 1 + 25.7T + 841T^{2} \) |
| 31 | \( 1 + 4.83iT - 961T^{2} \) |
| 37 | \( 1 + 35.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 55.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 104.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 49.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 72.9iT - 9.40e3T^{2} \) |
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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
−9.380620536628659218822839917262, −9.034186546953054054397902382600, −7.955258819355531445924822344520, −7.37042590825254962008061006821, −6.37888474328954080731467907041, −5.51082466040578691074028134644, −4.66775220740122549729852836956, −3.76537820862557536744962584254, −2.42027570267946773004746704333, −0.46027560854055808382078930367,
1.42105954312466276784817795128, 2.23658791972678349119278391019, 3.71144120383842610506650862551, 4.36809567455548036750251091784, 5.48324676535883296279812044092, 6.16927828929545303156986127534, 7.76322513119049312352361515465, 8.234993688987878234093304983277, 9.394623896765889115697387692811, 10.23026317944003887488619118166