| L(s) = 1 | + (−1.00 + 0.995i)2-s + (−0.799 − 1.53i)3-s + (0.0195 − 1.99i)4-s + (−0.411 − 2.19i)5-s + (2.33 + 0.748i)6-s − 3.34i·7-s + (1.97 + 2.02i)8-s + (−1.72 + 2.45i)9-s + (2.60 + 1.79i)10-s + (−5.09 + 3.70i)11-s + (−3.08 + 1.56i)12-s + (1.13 + 0.821i)13-s + (3.33 + 3.36i)14-s + (−3.04 + 2.39i)15-s + (−3.99 − 0.0782i)16-s + (−3.45 + 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.710 + 0.703i)2-s + (−0.461 − 0.887i)3-s + (0.00977 − 0.999i)4-s + (−0.184 − 0.982i)5-s + (0.952 + 0.305i)6-s − 1.26i·7-s + (0.696 + 0.717i)8-s + (−0.573 + 0.819i)9-s + (0.822 + 0.568i)10-s + (−1.53 + 1.11i)11-s + (−0.891 + 0.452i)12-s + (0.313 + 0.227i)13-s + (0.890 + 0.899i)14-s + (−0.786 + 0.617i)15-s + (−0.999 − 0.0195i)16-s + (−0.837 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0570627 - 0.334523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0570627 - 0.334523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.00 - 0.995i)T \) |
| 3 | \( 1 + (0.799 + 1.53i)T \) |
| 5 | \( 1 + (0.411 + 2.19i)T \) |
| good | 7 | \( 1 + 3.34iT - 7T^{2} \) |
| 11 | \( 1 + (5.09 - 3.70i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 0.821i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.45 - 1.12i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.47 + 0.479i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.31 + 3.86i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.46 + 0.801i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.51 - 1.14i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.63 + 3.37i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.595 + 0.819i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.65iT - 43T^{2} \) |
| 47 | \( 1 + (-1.36 + 4.19i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (8.76 + 2.84i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.57 + 4.05i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.88 + 1.36i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (11.2 - 3.64i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.51 + 4.64i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.90 + 5.01i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 3.67i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.69 + 5.20i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.68 - 7.81i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.29 - 7.06i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.99798354315668387550364865864, −10.51521937010010757231610607964, −9.245359507231243008164809569743, −8.169720633067765332504671532419, −7.44545134343361944696513337924, −6.77618676423610980114410992548, −5.34887965956395764429479411855, −4.59647611682048429289437306746, −1.84421010385698310903618267721, −0.31923962682081800337813114839,
2.70346983412646271982225804810, 3.37836147335579233762750517118, 5.12171039830244659896964905853, 6.15959891972118746559211248372, 7.61483479462750719164136371890, 8.680909957675010086709664336113, 9.426211498004504643862468886054, 10.55271033433439303096487441564, 11.07027023839423020491155831396, 11.63262037491776454329679162424
