L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.382 − 0.662i)5-s − 0.999i·8-s + (−0.662 − 0.382i)10-s − 1.84i·13-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s − 0.765·20-s + (0.207 − 0.358i)25-s + (−0.923 − 1.60i)26-s + (−0.866 − 0.499i)32-s + 1.84i·34-s + (0.707 + 1.22i)37-s + (−0.662 + 0.382i)40-s + 1.84·41-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.382 − 0.662i)5-s − 0.999i·8-s + (−0.662 − 0.382i)10-s − 1.84i·13-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s − 0.765·20-s + (0.207 − 0.358i)25-s + (−0.923 − 1.60i)26-s + (−0.866 − 0.499i)32-s + 1.84i·34-s + (0.707 + 1.22i)37-s + (−0.662 + 0.382i)40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.667505686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.667505686\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.84iT - T^{2} \) |
| 17 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 0.765iT - T^{2} \) |
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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
−9.376265564540767641262182437768, −8.331901937635687884491201020522, −7.83144316563523199415323965705, −6.54625041447735195470669254310, −5.89475190291607385504853077715, −5.01645283890148552296057764828, −4.26444073921001985255898916935, −3.39964242549572195257889258875, −2.36321983537022563119375797175, −0.986698763097903668707774752801,
2.14780905943798996472039927282, 2.99947708517848359050068665488, 4.14195188894714314177308490023, 4.62333128495802153515283665367, 5.75333818533394771824086213048, 6.70209376955859749358618792907, 7.10208679579773178768928915530, 7.80333123671907175471272098558, 9.026218481918852137830391613825, 9.406036253250504358086962126000