L(s) = 1 | + (−0.707 + 1.22i)3-s + (0.500 + 0.866i)9-s + (1 − 1.73i)11-s − 2.82·13-s + (2.12 − 3.67i)17-s + (2.12 + 3.67i)19-s + (4 + 6.92i)23-s + (2.5 − 4.33i)25-s − 5.65·27-s + 6·29-s + (−4.24 + 7.34i)31-s + (1.41 + 2.44i)33-s + (1 + 1.73i)37-s + (2.00 − 3.46i)39-s + 4.24·41-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.707i)3-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s − 0.784·13-s + (0.514 − 0.891i)17-s + (0.486 + 0.842i)19-s + (0.834 + 1.44i)23-s + (0.5 − 0.866i)25-s − 1.08·27-s + 1.11·29-s + (−0.762 + 1.31i)31-s + (0.246 + 0.426i)33-s + (0.164 + 0.284i)37-s + (0.320 − 0.554i)39-s + 0.662·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.357917953\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.357917953\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 3.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4.24 - 7.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.36 - 11.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 + 4.89i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−9.664904689586518034370091089874, −9.090441743191103931591203454583, −7.953605414352798993819291834461, −7.32007658392261651962143561349, −6.31953792528740537369119317346, −5.23870828171433511133869283646, −4.90069809233314858228064078108, −3.71513931728519856833913675379, −2.81473425437319622068627924319, −1.24692198063384022054149217095,
0.63319021045137542676633985441, 1.85369241102479446230580954752, 3.04358677556055609394986201667, 4.25759502492845363481291717683, 5.12036899283917017339053988644, 6.12314039423318498748921755265, 6.89415727070289315775410688960, 7.37510104018696288143274833276, 8.362439764050568671902491081988, 9.304332979260494770366589143817