L(s) = 1 | + (0.0713 + 0.997i)3-s + (0.281 + 0.959i)5-s + (0.479 − 0.877i)7-s + (−0.989 + 0.142i)9-s + (−0.959 − 0.281i)11-s + (−0.997 + 0.0713i)13-s + (−0.936 + 0.349i)15-s + (−0.909 + 0.415i)17-s + (−0.599 + 0.800i)19-s + (0.909 + 0.415i)21-s + (0.800 + 0.599i)23-s + (−0.841 + 0.540i)25-s + (−0.212 − 0.977i)27-s + (0.877 + 0.479i)29-s + (0.800 − 0.599i)31-s + ⋯ |
L(s) = 1 | + (0.0713 + 0.997i)3-s + (0.281 + 0.959i)5-s + (0.479 − 0.877i)7-s + (−0.989 + 0.142i)9-s + (−0.959 − 0.281i)11-s + (−0.997 + 0.0713i)13-s + (−0.936 + 0.349i)15-s + (−0.909 + 0.415i)17-s + (−0.599 + 0.800i)19-s + (0.909 + 0.415i)21-s + (0.800 + 0.599i)23-s + (−0.841 + 0.540i)25-s + (−0.212 − 0.977i)27-s + (0.877 + 0.479i)29-s + (0.800 − 0.599i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5764123022 - 0.2950417286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5764123022 - 0.2950417286i\) |
\(L(1)\) |
\(\approx\) |
\(0.8343634969 + 0.3151713194i\) |
\(L(1)\) |
\(\approx\) |
\(0.8343634969 + 0.3151713194i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.0713 + 0.997i)T \) |
| 5 | \( 1 + (0.281 + 0.959i)T \) |
| 7 | \( 1 + (0.479 - 0.877i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (-0.997 + 0.0713i)T \) |
| 17 | \( 1 + (-0.909 + 0.415i)T \) |
| 19 | \( 1 + (-0.599 + 0.800i)T \) |
| 23 | \( 1 + (0.800 + 0.599i)T \) |
| 29 | \( 1 + (0.877 + 0.479i)T \) |
| 31 | \( 1 + (0.800 - 0.599i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (-0.997 - 0.0713i)T \) |
| 43 | \( 1 + (0.877 - 0.479i)T \) |
| 47 | \( 1 + (0.755 - 0.654i)T \) |
| 53 | \( 1 + (0.755 + 0.654i)T \) |
| 59 | \( 1 + (-0.0713 + 0.997i)T \) |
| 61 | \( 1 + (-0.977 + 0.212i)T \) |
| 67 | \( 1 + (0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.281 - 0.959i)T \) |
| 73 | \( 1 + (0.142 - 0.989i)T \) |
| 79 | \( 1 + (0.989 + 0.142i)T \) |
| 83 | \( 1 + (-0.936 - 0.349i)T \) |
| 97 | \( 1 + (-0.959 + 0.281i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.588823632062467220194073550989, −21.5923669259533352494835498783, −20.84657562815378855424260955491, −20.049600114192273110878049000895, −19.240282979904655199948128978447, −18.43055510445853299449245029890, −17.46036992010228139867332361735, −17.290673375183915242997950673859, −15.84633897443242479271729598096, −15.18214286780609706442215895517, −14.096959673862014143226214089747, −13.248710644297764981415991591471, −12.57177313567651069419208555039, −12.00335121268319320547079243268, −11.01557132412589400940953255323, −9.751244786526407433973407586848, −8.68527931697315638260643182656, −8.32547851877004327162819039813, −7.198727785226691611908231756527, −6.28673534549274384839087041243, −5.07970287429966520695958802447, −4.78081737714944097465119554509, −2.610646082058074100246906561535, −2.293957765158177627567916454180, −0.95772353390762953481411838118,
0.15924030050503849601259788643, 2.06211250262560305618285932789, 2.98270780985739860305283880581, 3.983985859051454712043518593736, 4.86348846829179931391779543582, 5.81423783345329968531670949228, 6.961740836089914448346248876190, 7.81791655550426079401957655719, 8.82297467671694788443921700840, 9.97898217953223471579694497208, 10.57455925854034308524494368100, 10.96952486830729780882430403845, 12.147605189428535458130024638117, 13.5631163868578042817376013855, 14.02959178059279885196211897882, 15.07100099554405576795070531900, 15.3971800495820352698918597002, 16.692738131683148151631116271231, 17.26281709319872449225088576213, 18.04463607506995132897305839318, 19.18557837976299028075461495139, 19.854738984663975656346569619001, 20.94277305471448506209227972413, 21.36327017863507489420953356053, 22.1905215756745999419959154151