L(s) = 1 | + 4.60i·7-s − 3·9-s + 0.605i·11-s + 3.60·13-s − 7.21·17-s + 8.60i·19-s + 5·25-s + 7.21·29-s − 3.39i·31-s + 12.6i·47-s − 14.2·49-s − 2·53-s − 9.81i·59-s + 6·61-s − 13.8i·63-s + ⋯ |
L(s) = 1 | + 1.74i·7-s − 9-s + 0.182i·11-s + 1.00·13-s − 1.74·17-s + 1.97i·19-s + 25-s + 1.33·29-s − 0.609i·31-s + 1.83i·47-s − 2.03·49-s − 0.274·53-s − 1.27i·59-s + 0.768·61-s − 1.74i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796261 + 0.796261i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796261 + 0.796261i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - 3.60T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 0.605iT - 11T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 - 8.60iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 12.6iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 9.81iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 7.39iT - 67T^{2} \) |
| 71 | \( 1 + 5.81iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.48308413684132509939852473486, −10.70945337469870713271168118268, −9.390086620278790879119792793450, −8.649255121889692142486832193404, −8.139682816425469971211075865347, −6.34486026994241942371625440363, −5.93834191756031048581914741482, −4.73265220071857005222764546477, −3.18671270363421497151561504739, −2.06751429733452498427171523657,
0.72874106844732925400529890387, 2.79257035189951538222292282755, 4.05690055713297330648870858632, 5.00004897850268835114046377351, 6.58919352020252607552271392207, 6.98116411676419319043567600418, 8.446031963934661764694942469765, 8.930508368986412365801052672009, 10.36959634230983350571206324619, 10.98826745456527218978602120433