L(s) = 1 | + (0.707 − 1.22i)2-s + (0.707 + 1.22i)3-s + (0.792 − 1.37i)5-s + 2·6-s + 2.82·8-s + (0.500 − 0.866i)9-s + (−1.12 − 1.94i)10-s + (−2.12 − 3.67i)11-s + 13-s + 2.24·15-s + (2.00 − 3.46i)16-s + (0.707 + 1.22i)17-s + (−0.707 − 1.22i)18-s + (−3.62 + 6.27i)19-s − 6·22-s + (2.91 − 5.04i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)2-s + (0.408 + 0.707i)3-s + (0.354 − 0.614i)5-s + 0.816·6-s + 0.999·8-s + (0.166 − 0.288i)9-s + (−0.354 − 0.614i)10-s + (−0.639 − 1.10i)11-s + 0.277·13-s + 0.579·15-s + (0.500 − 0.866i)16-s + (0.171 + 0.297i)17-s + (−0.166 − 0.288i)18-s + (−0.830 + 1.43i)19-s − 1.27·22-s + (0.607 − 1.05i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.37339 - 0.994650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37339 - 0.994650i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.707 + 1.22i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.792 + 1.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.62 - 6.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.91 + 5.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 + (-1.62 - 2.80i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.12 - 1.94i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-0.792 + 1.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0857 - 0.148i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.171 - 0.297i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.24 - 12.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + (4.62 + 8.00i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.74 - 13.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + (-0.792 + 1.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.41457048761249141539298681594, −9.980815426560851673661177901669, −8.573885210882916808139095850815, −8.377430619927458432970466360627, −6.82551966833719843495439028913, −5.60749125144439519274871169167, −4.60961954635084684829247184784, −3.68573687606327661692587272626, −2.91914392144534114624932857818, −1.42615848981517564179753253240,
1.75699425223926330273685428828, 2.74592452961022508473951032622, 4.50928255514208587821403582372, 5.26962649302078517219834032259, 6.48714596614768755541098813799, 7.07062743328860484046445728327, 7.61929661693857340399963324565, 8.672935689509664228670251555336, 9.966694759796192967331010741763, 10.56156348783203605587083593668