L(s) = 1 | + (0.707 + 1.22i)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 − 2.82·8-s − 1.00i·9-s + 2i·11-s + (0.517 + 1.93i)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 + (1.22 − 0.707i)18-s − 7.19·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)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 − 0.999·8-s − 0.333i·9-s + 0.603i·11-s + (0.149 + 0.557i)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 + (0.288 − 0.166i)18-s − 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43365 + 1.18289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43365 + 1.18289i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 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
−12.03500718939647003412574885814, −11.45431611468202200783809683763, −9.751592232672664418176110646960, −8.708051383734173708209760536110, −8.146402906971523203432126502685, −6.99851954964959308171849961511, −6.15801581513089929057673338337, −4.93870917422369993105520452417, −3.82709420516811561868415359340, −2.19322927879178443119481888956,
1.40934423812448630895860693736, 3.20509406619514771883719907166, 4.01907419273732841686654262399, 5.23688680675532335963240735048, 6.30093874075467404149450173703, 8.124266647010406546289255559276, 8.640710963359188980863588637928, 10.07776759583067755214457455421, 10.70279637453085339646291086548, 11.23416275624515828710721085407