L(s) = 1 | + (−0.987 − 0.156i)2-s + (−0.809 + 0.587i)3-s + (0.951 + 0.309i)4-s + (0.891 − 0.453i)6-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)9-s + (−0.119 + 1.51i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)16-s + (0.581 − 0.497i)17-s + (−0.453 + 0.891i)18-s + (0.453 + 0.108i)19-s + (0.355 − 1.47i)22-s + (0.987 − 0.156i)24-s + (−0.587 + 0.809i)25-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.156i)2-s + (−0.809 + 0.587i)3-s + (0.951 + 0.309i)4-s + (0.891 − 0.453i)6-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)9-s + (−0.119 + 1.51i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)16-s + (0.581 − 0.497i)17-s + (−0.453 + 0.891i)18-s + (0.453 + 0.108i)19-s + (0.355 − 1.47i)22-s + (0.987 − 0.156i)24-s + (−0.587 + 0.809i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4704073405\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4704073405\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.453 - 0.891i)T \) |
good | 5 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (0.453 - 0.891i)T^{2} \) |
| 11 | \( 1 + (0.119 - 1.51i)T + (-0.987 - 0.156i)T^{2} \) |
| 13 | \( 1 + (-0.891 + 0.453i)T^{2} \) |
| 17 | \( 1 + (-0.581 + 0.497i)T + (0.156 - 0.987i)T^{2} \) |
| 19 | \( 1 + (-0.453 - 0.108i)T + (0.891 + 0.453i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.156 - 0.987i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \) |
| 47 | \( 1 + (-0.453 - 0.891i)T^{2} \) |
| 53 | \( 1 + (0.156 + 0.987i)T^{2} \) |
| 59 | \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 67 | \( 1 + (0.0123 + 0.156i)T + (-0.987 + 0.156i)T^{2} \) |
| 71 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 73 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - 0.618iT - T^{2} \) |
| 89 | \( 1 + (-1.70 + 1.04i)T + (0.453 - 0.891i)T^{2} \) |
| 97 | \( 1 + (-1.93 + 0.152i)T + (0.987 - 0.156i)T^{2} \) |
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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
−10.08022269104008197745180204152, −9.774077488856989782498240273196, −9.056447848483862789626909681961, −7.75671207214367847399218441155, −7.18436377470570792935872260346, −6.21976057469878928991425979102, −5.24884413129434228880270654084, −4.19754934138758644296517039493, −2.94735650782143123331770181642, −1.43552165517408539887573819544,
0.73450578042483804034159042729, 2.12336391843154232107267260688, 3.49919921949041773282994721990, 5.25441720700913866357364734906, 5.94140736432236681565781767813, 6.62858385556368961513542343756, 7.64505857505098322167505361623, 8.228147565122510719547466167948, 9.078359152559226807493770156316, 10.25766103104345018058726931705