L(s) = 1 | + (−1.40 + 0.178i)2-s + (−1.22 + 1.22i)3-s + (1.93 − 0.5i)4-s + (1.5 − 1.93i)6-s + (−2.62 + 1.04i)8-s − 2.99i·9-s + (−1.75 + 2.98i)12-s + (3.50 − 1.93i)16-s + (3.16 − 3.16i)17-s + (0.534 + 4.20i)18-s − 7.74·19-s + (−6.32 − 6.32i)23-s + (1.93 − 4.5i)24-s + (3.67 + 3.67i)27-s + 8·31-s + (−4.56 + 3.34i)32-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.126i)2-s + (−0.707 + 0.707i)3-s + (0.968 − 0.250i)4-s + (0.612 − 0.790i)6-s + (−0.929 + 0.370i)8-s − 0.999i·9-s + (−0.507 + 0.861i)12-s + (0.875 − 0.484i)16-s + (0.766 − 0.766i)17-s + (0.126 + 0.992i)18-s − 1.77·19-s + (−1.31 − 1.31i)23-s + (0.395 − 0.918i)24-s + (0.707 + 0.707i)27-s + 1.43·31-s + (−0.807 + 0.590i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445221 - 0.231773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445221 - 0.231773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.178i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-3.16 + 3.16i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 + (6.32 + 6.32i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (-6.32 + 6.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.79 + 9.79i)T - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 15.4iT - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + (-2.44 + 2.44i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−10.21607758252545900693765644183, −9.992933219152730934189956027246, −8.787514120811993037415024072382, −8.149954225411931243398296451494, −6.80425564756937079336153133721, −6.20974384915864853270670932564, −5.12278252603997158164036051832, −3.90254728416299566739404515771, −2.37155183435243092267282651382, −0.44790707596659336550707019144,
1.30493500508135058350400568727, 2.48009406165604311021459714368, 4.14180346667588527106241603210, 5.82604898544274820759791807010, 6.29015487804159195851256571599, 7.45336160691741824514279072337, 8.041721897272840499587544350109, 8.956476471775064143046051419102, 10.23627542386327430374573514117, 10.56926471524268426730606511369