L(s) = 1 | + i·2-s + i·3-s − 4-s + (−1.67 + 1.48i)5-s − 6-s + 3.35i·7-s − i·8-s − 9-s + (−1.48 − 1.67i)10-s − 0.962·11-s − i·12-s − 1.61i·13-s − 3.35·14-s + (−1.48 − 1.67i)15-s + 16-s + 0.387i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.749 + 0.662i)5-s − 0.408·6-s + 1.26i·7-s − 0.353i·8-s − 0.333·9-s + (−0.468 − 0.529i)10-s − 0.290·11-s − 0.288i·12-s − 0.447i·13-s − 0.895·14-s + (−0.382 − 0.432i)15-s + 0.250·16-s + 0.0940i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.242044 - 0.639135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.242044 - 0.639135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 3.35iT - 7T^{2} \) |
| 11 | \( 1 + 0.962T + 11T^{2} \) |
| 13 | \( 1 + 1.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.387iT - 17T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 1.61iT - 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 - 6.18iT - 43T^{2} \) |
| 47 | \( 1 + 0.962iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.22iT - 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 + 3.22iT - 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 + 4.64T + 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−11.23811577132475655377859608723, −10.33717637263800498103372671748, −9.411259596512340013290592119127, −8.482366464785332251064297780928, −7.87201827409224188344203101939, −6.71644791570147030826565811817, −5.79392936258933716759737369588, −4.91069135163245847719790275573, −3.71221800076445773810352542758, −2.62905359638996164785532931403,
0.38826299121997028302726111760, 1.73330909398726082051822632408, 3.43705856575655104629233546405, 4.27377215473536714194381455765, 5.30428127178339091425732015310, 6.80996599760350786897894865851, 7.61454655213406547320648486144, 8.390777369358195714726312157530, 9.358410936148320390042214983159, 10.36937190526628826602065150412