L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.32 − 1.11i)3-s − 1.00i·4-s + (−1.28 + 1.82i)5-s + (1.72 − 0.154i)6-s + (1.30 + 1.30i)7-s + (0.707 + 0.707i)8-s + (0.531 + 2.95i)9-s + (−0.384 − 2.20i)10-s − 5.15i·11-s + (−1.11 + 1.32i)12-s + (0.342 − 0.342i)13-s − 1.84·14-s + (3.74 − 1.00i)15-s − 1.00·16-s + (−4.25 + 4.25i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.767 − 0.641i)3-s − 0.500i·4-s + (−0.574 + 0.818i)5-s + (0.704 − 0.0628i)6-s + (0.493 + 0.493i)7-s + (0.250 + 0.250i)8-s + (0.177 + 0.984i)9-s + (−0.121 − 0.696i)10-s − 1.55i·11-s + (−0.320 + 0.383i)12-s + (0.0948 − 0.0948i)13-s − 0.493·14-s + (0.965 − 0.258i)15-s − 0.250·16-s + (−1.03 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00222615 - 0.148219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00222615 - 0.148219i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.32 + 1.11i)T \) |
| 5 | \( 1 + (1.28 - 1.82i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.15iT - 11T^{2} \) |
| 13 | \( 1 + (-0.342 + 0.342i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.25 - 4.25i)T - 17iT^{2} \) |
| 23 | \( 1 + (-3.32 - 3.32i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + (-0.317 - 0.317i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.89iT - 41T^{2} \) |
| 43 | \( 1 + (8.84 - 8.84i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.14 - 2.14i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.17 + 8.17i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 + (1.50 + 1.50i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 2.73i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.23iT - 79T^{2} \) |
| 83 | \( 1 + (-9.88 - 9.88i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + (-4.02 - 4.02i)T + 97iT^{2} \) |
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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
−11.22007969503999983332353279318, −10.69054826363325613198910195738, −9.242238067284220418833125080922, −8.243350520156987019154448276259, −7.68815846771993133802969864156, −6.59948255459293009151976791934, −6.01491766406405828448591877971, −5.02261041818851525760570877632, −3.42253982879995245195386183109, −1.77899572502617606595004104708,
0.10828399579358373558186295993, 1.77854682414320045655285250190, 3.77749638982296788774024434890, 4.55729901683102420762339889080, 5.26344618449509772039303731003, 6.99137651003287103234865448265, 7.53072225501612259289000029574, 8.999261667984268286886484129700, 9.307302691889436499044961437688, 10.45302925063636871164929103395