L(s) = 1 | + (1.22 + 1.22i)3-s + (3.67 − 3.67i)7-s + 2.99i·9-s + (1.22 + 1.22i)13-s − 7i·19-s + 9·21-s + (−3.67 + 3.67i)27-s + 11·31-s + (−4.89 + 4.89i)37-s + 2.99i·39-s + (−1.22 − 1.22i)43-s − 20i·49-s + (8.57 − 8.57i)57-s − 61-s + (11.0 + 11.0i)63-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (1.38 − 1.38i)7-s + 0.999i·9-s + (0.339 + 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (−0.707 + 0.707i)27-s + 1.97·31-s + (−0.805 + 0.805i)37-s + 0.480i·39-s + (−0.186 − 0.186i)43-s − 2.85i·49-s + (1.13 − 1.13i)57-s − 0.128·61-s + (1.38 + 1.38i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.488754915\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.488754915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.67 + 3.67i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1.22 - 1.22i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 11T + 31T^{2} \) |
| 37 | \( 1 + (4.89 - 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + (8.57 - 8.57i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-9.79 - 9.79i)T + 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (3.67 - 3.67i)T - 97iT^{2} \) |
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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
−9.883419415705688192408875294127, −8.797409752305538611306568622063, −8.256706726417695643030417323288, −7.44903624111486981578733417630, −6.66939362729089187202259762112, −5.05845320738339636101057930467, −4.57912296300235672015326611217, −3.77890601369810442353567788891, −2.54625793442581065754182216381, −1.21062990634165342564177201413,
1.42920356221139762698288822257, 2.25256315194500926898491817944, 3.32019184603891760414973137728, 4.59098462127085621477486890515, 5.66163786826726662510357345434, 6.29249271334468575033423357659, 7.58655141731098056298687027709, 8.200007308409463622674035543146, 8.617791554353112322188278304130, 9.506151683585505346193621786490