| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.826 − 1.52i)3-s + (0.939 − 0.342i)4-s + (1.99 + 1.67i)5-s + (1.07 + 1.35i)6-s + (0.187 + 2.63i)7-s + (−0.866 + 0.5i)8-s + (−1.63 + 2.51i)9-s + (−2.25 − 1.30i)10-s + (1.10 + 1.31i)11-s + (−1.29 − 1.14i)12-s + (−6.54 − 1.15i)13-s + (−0.642 − 2.56i)14-s + (0.902 − 4.42i)15-s + (0.766 − 0.642i)16-s + (−3.19 + 5.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.476 − 0.878i)3-s + (0.469 − 0.171i)4-s + (0.893 + 0.749i)5-s + (0.440 + 0.553i)6-s + (0.0708 + 0.997i)7-s + (−0.306 + 0.176i)8-s + (−0.545 + 0.838i)9-s + (−0.714 − 0.412i)10-s + (0.331 + 0.395i)11-s + (−0.374 − 0.331i)12-s + (−1.81 − 0.320i)13-s + (−0.171 − 0.685i)14-s + (0.232 − 1.14i)15-s + (0.191 − 0.160i)16-s + (−0.774 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628212 + 0.463573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628212 + 0.463573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (0.826 + 1.52i)T \) |
| 7 | \( 1 + (-0.187 - 2.63i)T \) |
| good | 5 | \( 1 + (-1.99 - 1.67i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 1.31i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (6.54 + 1.15i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.19 - 5.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.36 + 0.789i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.89 - 7.94i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.93 + 0.693i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.301 + 0.829i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.57 - 4.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.142 + 0.807i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.82 + 5.72i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.60 - 2.40i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 6.75iT - 53T^{2} \) |
| 59 | \( 1 + (0.690 + 0.579i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.46 + 4.02i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.06 - 11.7i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.69 - 1.55i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.04 - 0.605i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.88 + 10.6i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.170 + 0.965i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-1.37 - 2.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.7 - 14.0i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.57357013191494469572657694037, −10.53479602470944101438756137677, −9.762258518756718046549901063263, −8.814013515039924591700198937017, −7.64161082639157522391211970349, −6.86656477318530579284924834282, −6.00905908557772108916483598386, −5.16413913961951077621533400438, −2.65173508822893816141879813514, −1.87080833921555981866582428971,
0.67639405074914076458128712609, 2.64943405385371455047193683174, 4.42458729059959793215410260178, 5.11492887929862363166002908936, 6.47383810960217579475558973574, 7.38804562295204897739135047864, 8.911243022041109000051506746235, 9.379949199582151329861695411470, 10.17410103907703076144114225525, 10.89110200588417528924838273105
