L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.707 + 0.707i)3-s + (0.999 + 1.73i)4-s + (1.36 − 0.366i)6-s + (−1.74 − 1.74i)7-s − 2.82i·8-s − 1.00i·9-s − 2i·11-s + (−1.93 − 0.517i)12-s + (4.05 + 4.05i)13-s + (0.901 + 3.36i)14-s + (−2.00 + 3.46i)16-s + (4.24 − 4.24i)17-s + (−0.707 + 1.22i)18-s + 7.19·19-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (−0.408 + 0.408i)3-s + (0.499 + 0.866i)4-s + (0.557 − 0.149i)6-s + (−0.658 − 0.658i)7-s − 0.999i·8-s − 0.333i·9-s − 0.603i·11-s + (−0.557 − 0.149i)12-s + (1.12 + 1.12i)13-s + (0.241 + 0.899i)14-s + (−0.500 + 0.866i)16-s + (1.02 − 1.02i)17-s + (−0.166 + 0.288i)18-s + 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.704284 - 0.262963i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.704284 - 0.262963i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.74 + 1.74i)T + 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-4.05 - 4.05i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + (0.378 - 0.378i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.46iT - 29T^{2} \) |
| 31 | \( 1 + 0.267iT - 31T^{2} \) |
| 37 | \( 1 + (-2.07 + 2.07i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + (-1.74 + 1.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.14 - 9.14i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.03 - 1.03i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.53T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + (8.81 + 8.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.46iT - 71T^{2} \) |
| 73 | \( 1 + (2.82 + 2.82i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 + (0.656 - 0.656i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (4.43 - 4.43i)T - 97iT^{2} \) |
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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
−11.53134728088508847738641066820, −10.66915712388318635495114202680, −9.685764237209467808405558334846, −9.203690835129743405870333414419, −7.86613859997766633086490789798, −6.91636847323786099669323612358, −5.82108354802297532821676286716, −4.06911851531989675583786129916, −3.12776389750713360144645410640, −0.953076200547883476621943553750,
1.30192589175606002282227612411, 3.15591942293845268274111660768, 5.39966334786721538441851554985, 5.94631669678817738058631871140, 7.08075961985905466094740085119, 7.993691309437563466282345128106, 8.917929458032088371801583673515, 9.995744338858016055722009257712, 10.65104952117280602504545143032, 11.82925061662946232355446989120