L(s) = 1 | + (−0.738 − 0.380i)2-s + (−0.179 − 0.252i)4-s + (0.154 + 1.07i)8-s + (0.235 + 0.971i)9-s + (0.252 − 0.130i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 0.236·22-s + (0.723 + 0.690i)23-s + (0.0475 + 0.998i)25-s + (0.698 + 1.53i)29-s + (0.430 − 0.410i)32-s + (0.202 − 0.234i)36-s + (−0.308 − 1.27i)37-s + (0.273 − 1.89i)43-s + (−0.0783 − 0.0403i)44-s + ⋯ |
L(s) = 1 | + (−0.738 − 0.380i)2-s + (−0.179 − 0.252i)4-s + (0.154 + 1.07i)8-s + (0.235 + 0.971i)9-s + (0.252 − 0.130i)11-s + (0.194 − 0.561i)16-s + (0.195 − 0.807i)18-s − 0.236·22-s + (0.723 + 0.690i)23-s + (0.0475 + 0.998i)25-s + (0.698 + 1.53i)29-s + (0.430 − 0.410i)32-s + (0.202 − 0.234i)36-s + (−0.308 − 1.27i)37-s + (0.273 − 1.89i)43-s + (−0.0783 − 0.0403i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6712512587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6712512587\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (-0.723 - 0.690i)T \) |
good | 2 | \( 1 + (0.738 + 0.380i)T + (0.580 + 0.814i)T^{2} \) |
| 3 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 5 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 11 | \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 19 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 29 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 37 | \( 1 + (0.308 + 1.27i)T + (-0.888 + 0.458i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.65 + 0.318i)T + (0.928 - 0.371i)T^{2} \) |
| 59 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 61 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 67 | \( 1 + (-0.0395 - 0.829i)T + (-0.995 + 0.0950i)T^{2} \) |
| 71 | \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 79 | \( 1 + (-1.65 - 0.318i)T + (0.928 + 0.371i)T^{2} \) |
| 83 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 97 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
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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
−10.11256937203840080270559135919, −9.082020925889428223296199721284, −8.716474858136771271874367813718, −7.64022068575490872451377084514, −6.95177100783657225194328455898, −5.49187700672356259485614700708, −5.09268317544009278754720871280, −3.77403182196438560358584001858, −2.39738862172697916757190818073, −1.31878387507843658490335213366,
0.942779325522228085969958616301, 2.80320191212524938330634523527, 3.94722689495707149123745007492, 4.69927221062562924299781407412, 6.29037273125573665969544518916, 6.66340964109259009335116757566, 7.72428407189936652001874211047, 8.411225614055569304410324510037, 9.174354695934854195359296965648, 9.814413964265847423765880114005