L(s) = 1 | + (0.956 + 1.44i)3-s + (2.60 + 0.477i)7-s + (−1.16 + 2.76i)9-s + (1.34 − 0.773i)11-s + 4.18i·13-s + (2.55 + 4.42i)17-s + (−4.62 − 2.67i)19-s + (1.80 + 4.21i)21-s + (−3.15 − 1.82i)23-s + (−5.10 + 0.954i)27-s + 9.79i·29-s + (6.79 − 3.92i)31-s + (2.39 + 1.19i)33-s + (1.71 − 2.96i)37-s + (−6.04 + 4.00i)39-s + ⋯ |
L(s) = 1 | + (0.552 + 0.833i)3-s + (0.983 + 0.180i)7-s + (−0.389 + 0.920i)9-s + (0.404 − 0.233i)11-s + 1.16i·13-s + (0.619 + 1.07i)17-s + (−1.06 − 0.613i)19-s + (0.392 + 0.919i)21-s + (−0.658 − 0.380i)23-s + (−0.982 + 0.183i)27-s + 1.81i·29-s + (1.22 − 0.705i)31-s + (0.417 + 0.208i)33-s + (0.281 − 0.488i)37-s + (−0.967 + 0.641i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.257607534\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.257607534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.956 - 1.44i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.60 - 0.477i)T \) |
good | 11 | \( 1 + (-1.34 + 0.773i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.18iT - 13T^{2} \) |
| 17 | \( 1 + (-2.55 - 4.42i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.79iT - 29T^{2} \) |
| 31 | \( 1 + (-6.79 + 3.92i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.71 + 2.96i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 + (-1.24 + 2.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.684 + 0.395i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.73 - 4.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.76 + 3.90i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.58 - 9.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.97iT - 71T^{2} \) |
| 73 | \( 1 + (-10.7 + 6.19i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.12 + 1.94i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + (7.65 - 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−9.164548164816649381067894415921, −8.566273932710746955514456548075, −8.142186740967712984923081197416, −7.04404788133480083183052436618, −6.12089417710344036900293113960, −5.14713874133240495375719181067, −4.34660723158027512873621454112, −3.78822212725364750896475550366, −2.49728034415149156509040858890, −1.60348743370820589863201234218,
0.75470352214213315819163044801, 1.85045239417783377126914173160, 2.80643714687744983081723687639, 3.84741257161573831046974339310, 4.83919171959186898733303468203, 5.83438653254236880760564764912, 6.58643845077686328031774014642, 7.59574313161713740658201662126, 8.017780287417440388400653447449, 8.565713054407043445087900229936