| L(s) = 1 | − 1.41i·2-s + (1 + 1.41i)3-s − 2.00·4-s + (2.00 − 1.41i)6-s + 2.82i·8-s + (−1.00 + 2.82i)9-s + 2.82i·11-s + (−2.00 − 2.82i)12-s + 4.00·16-s + 5.65i·17-s + (4.00 + 1.41i)18-s + 2·19-s + 4.00·22-s + (−4.00 + 2.82i)24-s + (−5.00 + 1.41i)27-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s + (0.577 + 0.816i)3-s − 1.00·4-s + (0.816 − 0.577i)6-s + 1.00i·8-s + (−0.333 + 0.942i)9-s + 0.852i·11-s + (−0.577 − 0.816i)12-s + 1.00·16-s + 1.37i·17-s + (0.942 + 0.333i)18-s + 0.458·19-s + 0.852·22-s + (−0.816 + 0.577i)24-s + (−0.962 + 0.272i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37190 + 0.436043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37190 + 0.436043i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 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
−10.61595755265602291452998266270, −9.953455076161035232417059273905, −9.266521253316885301013931231556, −8.404478465358880357456116114118, −7.57678308759711453796633563936, −5.89936961037098458437174484953, −4.75612780775850879758142922509, −4.02659878442431253323237839405, −2.97620352145636067292891802234, −1.79127616560145264208019356309,
0.796486668414836834978115388766, 2.78019077983702519044124188914, 3.94744695594640439918965092514, 5.35628999070245721891993393281, 6.16215486645643186765499032966, 7.23910810630651099409080134120, 7.65169802598765604280542852865, 8.862850676058498335260499243325, 9.129111699901980850780422569453, 10.38372129917558441687791126916
