L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 5-s + 4.24i·7-s − 2.82i·8-s + 1.41i·10-s − 1.41i·11-s + 4.24i·13-s − 6·14-s + 4.00·16-s + 2.82i·17-s − 4·19-s − 2.00·20-s + 2.00·22-s − 6·23-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + 0.447·5-s + 1.60i·7-s − 1.00i·8-s + 0.447i·10-s − 0.426i·11-s + 1.17i·13-s − 1.60·14-s + 1.00·16-s + 0.685i·17-s − 0.917·19-s − 0.447·20-s + 0.426·22-s − 1.25·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.348192 + 1.09550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.348192 + 1.09550i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 1.41iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−12.19586805110433613156463045948, −10.76146615434163062030132237743, −9.629742436404408414480087853254, −8.765284634998219099644236623898, −8.356782683169749804271540209582, −6.79966636559695846563248660184, −6.06441519930472116052374323927, −5.27898452220307407361595080133, −3.97855209757281879606308267309, −2.18297634845730494833523071809,
0.820602284773081203748656317357, 2.49963501840937754563527705343, 3.89033460147731916200278076854, 4.74298174087581268242913000961, 6.11315679310145380872364391788, 7.49788128753338031647520693614, 8.321898369333204782780546317636, 9.732161456972398700418152816844, 10.18613104174819670831886672943, 10.88435057944804444712052911222