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 + (−0.907 + 1.75i)11-s + (0.194 + 0.561i)16-s + (0.195 + 0.807i)18-s + 1.64i·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 + (1.46 + 0.356i)37-s + (−0.557 + 0.0801i)43-s + (−0.280 − 0.544i)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 + (−0.907 + 1.75i)11-s + (0.194 + 0.561i)16-s + (0.195 + 0.807i)18-s + 1.64i·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 + (1.46 + 0.356i)37-s + (−0.557 + 0.0801i)43-s + (−0.280 − 0.544i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.253968726\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253968726\) |
\(L(1)\) |
|
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 | \( 1 + (0.907 - 1.75i)T + (-0.580 - 0.814i)T^{2} \) |
| 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 | \( 1 + (-1.46 - 0.356i)T + (0.888 + 0.458i)T^{2} \) |
| 41 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.557 - 0.0801i)T + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.204 - 1.06i)T + (-0.928 - 0.371i)T^{2} \) |
| 59 | \( 1 + (-0.786 - 0.618i)T^{2} \) |
| 61 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 67 | \( 1 + (-1.81 - 0.0865i)T + (0.995 + 0.0950i)T^{2} \) |
| 71 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (-0.327 - 0.945i)T^{2} \) |
| 79 | \( 1 + (-0.204 - 1.06i)T + (-0.928 + 0.371i)T^{2} \) |
| 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
−10.25566838775057487407405670849, −9.489574345803290473391620687430, −8.194373256464341695127619989657, −7.918184340132845762145890023447, −6.82018034682217117822321887226, −5.59738546103247481850332982423, −4.65798225039451442870314602116, −4.37546192976392089483737882341, −2.73690944517569438473970880923, −2.24708366591360727879392779405,
0.937083859639225685074738058789, 3.10136700277908665288332223359, 3.61202070709766783918558877505, 4.97836497181636394276796795504, 5.60335935705367149499514790567, 6.31836657176755158324767837856, 7.17050313436625996890157640314, 8.333933628317557442319651471928, 9.057800703931550667652796117045, 9.805347852884079644482188787516