L(s) = 1 | + (1.86 − 1.86i)2-s − 4.93i·4-s + (−1.50 − 1.65i)5-s + (1.46 + 2.20i)7-s + (−5.45 − 5.45i)8-s + (−5.88 − 0.272i)10-s + 1.46·11-s + (−0.887 + 0.887i)13-s + (6.82 + 1.38i)14-s − 10.4·16-s + (−2.10 − 2.10i)17-s + 3.95·19-s + (−8.14 + 7.42i)20-s + (2.72 − 2.72i)22-s + (4.13 + 4.13i)23-s + ⋯ |
L(s) = 1 | + (1.31 − 1.31i)2-s − 2.46i·4-s + (−0.673 − 0.739i)5-s + (0.552 + 0.833i)7-s + (−1.92 − 1.92i)8-s + (−1.85 − 0.0862i)10-s + 0.441·11-s + (−0.246 + 0.246i)13-s + (1.82 + 0.370i)14-s − 2.61·16-s + (−0.510 − 0.510i)17-s + 0.908·19-s + (−1.82 + 1.66i)20-s + (0.580 − 0.580i)22-s + (0.861 + 0.861i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.878940 - 2.16719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.878940 - 2.16719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.50 + 1.65i)T \) |
| 7 | \( 1 + (-1.46 - 2.20i)T \) |
good | 2 | \( 1 + (-1.86 + 1.86i)T - 2iT^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + (0.887 - 0.887i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.10 + 2.10i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + (-4.13 - 4.13i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.18iT - 29T^{2} \) |
| 31 | \( 1 + 6.10iT - 31T^{2} \) |
| 37 | \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.769iT - 41T^{2} \) |
| 43 | \( 1 + (5.18 + 5.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.57 - 8.57i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.544 - 0.544i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 - 1.42iT - 61T^{2} \) |
| 67 | \( 1 + (5.93 - 5.93i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 + (6.81 - 6.81i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.52iT - 79T^{2} \) |
| 83 | \( 1 + (6.75 - 6.75i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + (-8.68 - 8.68i)T + 97iT^{2} \) |
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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
−11.62883041820573325471117756241, −10.95514344546520241162736515193, −9.522277061107717951100677761012, −8.904741489254686335807316630306, −7.33581964943913813714774306507, −5.72077563233768575787658125317, −4.98029228481635652516381955080, −4.08612448238047100063171536641, −2.83687800586061885755554196393, −1.38595822707340274749855971369,
3.07369463881918122498341207968, 4.10666712069369793881875330038, 4.90794204553580621524577965629, 6.28224009133700522274188870184, 7.06322505034408472492430055027, 7.72815979836781930681085516741, 8.634917267862497602099855869755, 10.38469006458678977926192617377, 11.43246456528422862224123091383, 12.15868866907023746669087525967