L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.25 + 0.368i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 1.29i)6-s + (0.415 + 0.909i)8-s + (0.601 + 0.386i)9-s + (0.345 − 0.755i)11-s − 1.30·12-s + (0.841 − 0.540i)16-s + (0.0405 − 0.281i)17-s + (0.297 − 0.650i)18-s + (0.698 + 0.449i)19-s + (−0.797 − 0.234i)22-s + (0.186 + 1.29i)24-s + (−0.654 + 0.755i)25-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.25 + 0.368i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 1.29i)6-s + (0.415 + 0.909i)8-s + (0.601 + 0.386i)9-s + (0.345 − 0.755i)11-s − 1.30·12-s + (0.841 − 0.540i)16-s + (0.0405 − 0.281i)17-s + (0.297 − 0.650i)18-s + (0.698 + 0.449i)19-s + (−0.797 − 0.234i)22-s + (0.186 + 1.29i)24-s + (−0.654 + 0.755i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.190010646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.190010646\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
good | 3 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 7 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.91 - 0.563i)T + (0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 83 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
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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
−10.29829982623371478640165043711, −9.629697591498970878286645030122, −8.947200210867108794053825681606, −8.276232896255895890597842077080, −7.46824394133979216320472151609, −5.86530410198500265734434016369, −4.64861233789756483920775232919, −3.52056049804484945678933927214, −3.05212854071229786676559595183, −1.67365923784166771188416425982,
1.80711825186729283861718714979, 3.29016378712050823822261355823, 4.35204473562697950408046640380, 5.47372403399733601554554735782, 6.69307085510735417624787415857, 7.34967866036938906629110355826, 8.162834018412822838870528886126, 8.799998435442587017513476209521, 9.598083749700947419735927157955, 10.28897978433208613963442476255