L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.67 + 0.448i)3-s + 1.00i·4-s + (1.5 + 0.866i)6-s + (2.44 − 2.44i)7-s + (0.707 − 0.707i)8-s + (2.59 − 1.50i)9-s − 5.19i·11-s + (−0.448 − 1.67i)12-s − 3.46·14-s − 1.00·16-s + (2.12 + 2.12i)17-s + (−2.89 − 0.776i)18-s + i·19-s + (−3 + 5.19i)21-s + (−3.67 + 3.67i)22-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.965 + 0.258i)3-s + 0.500i·4-s + (0.612 + 0.353i)6-s + (0.925 − 0.925i)7-s + (0.250 − 0.250i)8-s + (0.866 − 0.5i)9-s − 1.56i·11-s + (−0.129 − 0.482i)12-s − 0.925·14-s − 0.250·16-s + (0.514 + 0.514i)17-s + (−0.683 − 0.183i)18-s + 0.229i·19-s + (−0.654 + 1.13i)21-s + (−0.783 + 0.783i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.587230 - 0.392270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.587230 - 0.392270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (2.44 - 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (2.44 + 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (4.24 - 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (6.12 + 6.12i)T + 73iT^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 + (2.12 - 2.12i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \) |
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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
−12.58210027572103248218168073147, −11.42848597944046326395313119215, −10.90440794372993143752989206746, −10.16701299329317054463909194924, −8.734528717879513437390917410713, −7.67731401980368120566367480056, −6.33333379074247857526087454007, −4.94559894868430505510387522605, −3.61963447891937566463349296429, −1.05289566278120391432237533337,
1.82213642098685371348147253708, 4.80060521669202178668491220590, 5.51721419258324170294415894852, 6.94871938831552406350417645837, 7.72309903438764644110754356374, 9.111891318796405756136111000776, 10.11553632666089047855197758032, 11.29327025627552001909841518113, 12.02825674817194661126547498483, 13.00067056543088214102887937930