L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.698 + 1.53i)3-s + (−0.142 − 0.989i)4-s + (−1.61 − 0.474i)6-s + (0.841 + 0.540i)8-s + (−1.19 + 1.38i)9-s + (−1.61 + 0.474i)11-s + (1.41 − 0.909i)12-s + (−0.959 + 0.281i)16-s + (0.0405 − 0.281i)17-s + (−0.260 − 1.81i)18-s + (1.25 + 1.45i)19-s + (0.698 − 1.53i)22-s + (−0.239 + 1.66i)24-s + (0.841 − 0.540i)25-s + ⋯ |
L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.698 + 1.53i)3-s + (−0.142 − 0.989i)4-s + (−1.61 − 0.474i)6-s + (0.841 + 0.540i)8-s + (−1.19 + 1.38i)9-s + (−1.61 + 0.474i)11-s + (1.41 − 0.909i)12-s + (−0.959 + 0.281i)16-s + (0.0405 − 0.281i)17-s + (−0.260 − 1.81i)18-s + (1.25 + 1.45i)19-s + (0.698 − 1.53i)22-s + (−0.239 + 1.66i)24-s + (0.841 − 0.540i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7360446957\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7360446957\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
good | 3 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \) |
| 13 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 17 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 23 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 43 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 53 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 83 | \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + 0.284T + T^{2} \) |
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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
−10.64799543252047697629602903193, −10.33455272038624175126956215063, −9.655961054913502501190805377440, −8.767053031231123981472762061164, −8.028608109590587805216572591386, −7.21231233638758520413216725506, −5.52484798039744859195897799776, −5.08282791755478151274583780175, −3.84427770870222337577330776376, −2.47055069026038840425819143713,
1.11566733756708459161912107450, 2.60452187156297139317554090367, 3.09018930093223900587762207983, 5.00960081716739590539091804079, 6.52002486212915778765982053823, 7.54084027094721985404506548697, 7.922530497650540089478569684273, 8.818329787515971685407546547040, 9.654886033671284524077877546089, 10.88151632055067000890875081289