L(s) = 1 | + (−0.391 − 0.919i)2-s + (−0.692 + 0.721i)4-s + (0.987 + 0.160i)5-s + (0.935 + 0.354i)8-s + (−0.600 + 0.799i)9-s + (−0.239 − 0.970i)10-s + (0.632 − 0.774i)13-s + (−0.0402 − 0.999i)16-s + (−0.201 + 1.98i)17-s + (0.970 + 0.239i)18-s + (−0.799 + 0.600i)20-s + (0.948 + 0.316i)25-s + (−0.960 − 0.278i)26-s + (−0.156 − 0.368i)29-s + (−0.903 + 0.428i)32-s + ⋯ |
L(s) = 1 | + (−0.391 − 0.919i)2-s + (−0.692 + 0.721i)4-s + (0.987 + 0.160i)5-s + (0.935 + 0.354i)8-s + (−0.600 + 0.799i)9-s + (−0.239 − 0.970i)10-s + (0.632 − 0.774i)13-s + (−0.0402 − 0.999i)16-s + (−0.201 + 1.98i)17-s + (0.970 + 0.239i)18-s + (−0.799 + 0.600i)20-s + (0.948 + 0.316i)25-s + (−0.960 − 0.278i)26-s + (−0.156 − 0.368i)29-s + (−0.903 + 0.428i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.099423275\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099423275\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.391 + 0.919i)T \) |
| 5 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (-0.632 + 0.774i)T \) |
good | 3 | \( 1 + (0.600 - 0.799i)T^{2} \) |
| 7 | \( 1 + (0.200 - 0.979i)T^{2} \) |
| 11 | \( 1 + (-0.960 - 0.278i)T^{2} \) |
| 17 | \( 1 + (0.201 - 1.98i)T + (-0.979 - 0.200i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.156 + 0.368i)T + (-0.692 + 0.721i)T^{2} \) |
| 31 | \( 1 + (-0.935 - 0.354i)T^{2} \) |
| 37 | \( 1 + (0.458 - 0.965i)T + (-0.632 - 0.774i)T^{2} \) |
| 41 | \( 1 + (-1.65 + 0.828i)T + (0.600 - 0.799i)T^{2} \) |
| 43 | \( 1 + (-0.774 - 0.632i)T^{2} \) |
| 47 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 53 | \( 1 + (1.81 - 0.816i)T + (0.663 - 0.748i)T^{2} \) |
| 59 | \( 1 + (0.0804 - 0.996i)T^{2} \) |
| 61 | \( 1 + (-0.625 - 0.101i)T + (0.948 + 0.316i)T^{2} \) |
| 67 | \( 1 + (-0.0402 + 0.999i)T^{2} \) |
| 71 | \( 1 + (-0.391 - 0.919i)T^{2} \) |
| 73 | \( 1 + (-0.992 + 0.120i)T + (0.970 - 0.239i)T^{2} \) |
| 79 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 83 | \( 1 + (-0.120 - 0.992i)T^{2} \) |
| 89 | \( 1 + (-1.84 - 0.494i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.879 + 1.39i)T + (-0.428 + 0.903i)T^{2} \) |
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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
−8.861561383956638912536186621747, −8.201350316881622159720143535614, −7.68343142223231952605789769957, −6.35962011016613396384366370286, −5.77999989555458136866766643763, −4.93384518219297124371300112444, −3.92730818775265882672454154587, −3.01489237013391149154592345658, −2.19139477113243849304379476229, −1.35287315897692800977886699641,
0.819892998458519891692189359340, 2.07920935873650798175884486284, 3.28050056623115910052690718954, 4.46245710492250967874035800122, 5.22832222616434326944196337740, 5.90060928862336183456594036735, 6.61755908948718679194475067032, 7.08818093510706958869035128558, 8.143340792761199738415202942132, 8.917050482709054726056806826725