L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.5 + 0.866i)5-s + (1.62 + 2.09i)7-s + 2.82·8-s + (0.707 + 1.22i)10-s + (1.70 + 2.95i)11-s − 1.58·13-s + (3.70 − 0.507i)14-s + (2.00 − 3.46i)16-s + (−3.12 − 5.40i)17-s + (3.32 − 5.76i)19-s + 4.82·22-s + (−3.12 + 5.40i)23-s + (−0.499 − 0.866i)25-s + (−1.12 + 1.94i)26-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)2-s + (−0.223 + 0.387i)5-s + (0.612 + 0.790i)7-s + 0.999·8-s + (0.223 + 0.387i)10-s + (0.514 + 0.891i)11-s − 0.439·13-s + (0.990 − 0.135i)14-s + (0.500 − 0.866i)16-s + (−0.757 − 1.31i)17-s + (0.763 − 1.32i)19-s + 1.02·22-s + (−0.650 + 1.12i)23-s + (−0.0999 − 0.173i)25-s + (−0.219 + 0.380i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84001 - 0.300926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84001 - 0.300926i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 2 | \( 1 + (-0.707 + 1.22i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 2.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + (3.12 + 5.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.242T + 29T^{2} \) |
| 31 | \( 1 + (-0.0857 - 0.148i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.79 + 4.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (-4.65 + 8.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.585 - 1.01i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.86 + 10.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 + (1.03 + 1.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.32 + 4.03i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 + (1.87 - 3.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−11.75225051268908003480249341297, −11.10514374552009644547166763433, −9.875149719136042831238414980973, −9.007065153136923725670233193022, −7.59800947998285000878560311674, −6.96706377875793270950840113730, −5.22513359482241011609849043808, −4.39247059935920375892407219364, −2.98404141632036521417377373953, −1.98947451530844239465567255704,
1.48496743719243089836090257714, 3.84468998392254083656486381308, 4.69328974760513912708376762410, 5.87676115550160208194034638682, 6.69030570590213875083048281646, 7.891052026420932181237098485340, 8.416777543497311114334377257040, 9.994629222192400025889297957654, 10.77196273631920059902294916584, 11.71164066448348379248031098438