L(s) = 1 | + (0.309 − 0.951i)3-s + (−1.49 − 1.66i)5-s − 4.78·7-s + (−0.809 − 0.587i)9-s + (−1.58 + 1.14i)11-s + (−0.873 − 0.634i)13-s + (−2.04 + 0.909i)15-s + (−1.17 − 3.61i)17-s + (1.31 + 4.04i)19-s + (−1.47 + 4.54i)21-s + (4.74 − 3.44i)23-s + (−0.522 + 4.97i)25-s + (−0.809 + 0.587i)27-s + (3.26 − 10.0i)29-s + (−1.33 − 4.10i)31-s + ⋯ |
L(s) = 1 | + (0.178 − 0.549i)3-s + (−0.669 − 0.743i)5-s − 1.80·7-s + (−0.269 − 0.195i)9-s + (−0.477 + 0.346i)11-s + (−0.242 − 0.176i)13-s + (−0.527 + 0.234i)15-s + (−0.285 − 0.877i)17-s + (0.301 + 0.927i)19-s + (−0.322 + 0.992i)21-s + (0.989 − 0.718i)23-s + (−0.104 + 0.994i)25-s + (−0.155 + 0.113i)27-s + (0.605 − 1.86i)29-s + (−0.239 − 0.737i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0877302 - 0.518692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0877302 - 0.518692i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (1.49 + 1.66i)T \) |
good | 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 + (1.58 - 1.14i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.873 + 0.634i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.17 + 3.61i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.31 - 4.04i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.74 + 3.44i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.26 + 10.0i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.33 + 4.10i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.57 + 3.32i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.694 + 0.504i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + (-0.927 + 2.85i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.30 + 4.01i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.85 - 2.80i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.93 - 2.13i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.14 - 6.59i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-13.7 + 9.96i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.04 - 6.29i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.797 + 2.45i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.673 + 0.489i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.81 - 8.67i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.65942838157852081597909747160, −10.16702317853417903133627195062, −9.446904002261978257724595169963, −8.434163247634725654815031199060, −7.40847736745945093360533869768, −6.55856735290498156648593683294, −5.32080992557715849671829713460, −3.87718474866200714560064479084, −2.67694257915881551962844570464, −0.35959520115248816312758783364,
3.01848874051742557511064495197, 3.48591427016371792279880665852, 5.05498044443543253267349679617, 6.50832075327443187932744727582, 7.11028056441972544991683686747, 8.524366915266454417040662452888, 9.404777793254738392109839011354, 10.38327248840522999012006097078, 10.96409919077239009178156278455, 12.17909928545689529440090578852