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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0197615 - 0.0994116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0197615 - 0.0994116i\) |
\(L(\frac{3}{2})\) |
|
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
−12.29192440528789444229604017585, −11.26745644768735622696166120472, −10.26255779678570907211582791729, −9.123704856185168272258590071244, −8.636132729265639890776416227880, −7.11228189688458920285420090376, −6.65491029584522717293371731899, −5.56987924168041338695389283063, −4.47427741858996139811691191966, −2.25556464027707890346442398374,
0.085887207855476897652543601301, 2.33696543314545808332484879466, 3.84293031434998627054576700185, 4.70488089812656058169457790238, 6.34309904609486027778129801925, 7.49533897387227842140351548008, 8.562010005452052408090871919457, 9.707586722525360959071960892808, 10.34483261619798458985501369462, 10.88933254014640328776370847802