L(s) = 1 | + (0.900 + 0.433i)2-s + (−0.0747 + 0.997i)3-s + (0.623 + 0.781i)4-s + (−0.955 − 0.294i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.222 + 0.974i)8-s + (−0.988 − 0.149i)9-s + (−0.733 − 0.680i)10-s + (0.623 − 0.781i)11-s + (−0.826 + 0.563i)12-s + (−0.733 + 0.680i)13-s + (0.0747 + 0.997i)14-s + (0.365 − 0.930i)15-s + (−0.222 + 0.974i)16-s + (0.955 − 0.294i)17-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)2-s + (−0.0747 + 0.997i)3-s + (0.623 + 0.781i)4-s + (−0.955 − 0.294i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.222 + 0.974i)8-s + (−0.988 − 0.149i)9-s + (−0.733 − 0.680i)10-s + (0.623 − 0.781i)11-s + (−0.826 + 0.563i)12-s + (−0.733 + 0.680i)13-s + (0.0747 + 0.997i)14-s + (0.365 − 0.930i)15-s + (−0.222 + 0.974i)16-s + (0.955 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.055024622 + 1.835593959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055024622 + 1.835593959i\) |
\(L(1)\) |
\(\approx\) |
\(1.229810082 + 0.9898779194i\) |
\(L(1)\) |
\(\approx\) |
\(1.229810082 + 0.9898779194i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 3 | \( 1 + (-0.0747 + 0.997i)T \) |
| 5 | \( 1 + (-0.955 - 0.294i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.733 + 0.680i)T \) |
| 17 | \( 1 + (0.955 - 0.294i)T \) |
| 19 | \( 1 + (0.988 - 0.149i)T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 29 | \( 1 + (-0.0747 - 0.997i)T \) |
| 31 | \( 1 + (0.826 - 0.563i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.733 - 0.680i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (-0.826 - 0.563i)T \) |
| 67 | \( 1 + (-0.988 + 0.149i)T \) |
| 71 | \( 1 + (-0.365 + 0.930i)T \) |
| 73 | \( 1 + (0.733 - 0.680i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.0747 - 0.997i)T \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T \) |
| 97 | \( 1 + (0.623 - 0.781i)T \) |
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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
−33.97550135264056635260129101293, −32.60451954721551336237255209603, −31.24027129156225389800731709504, −30.40825038527799884094897781277, −29.81676953630929058345173397766, −28.36266031607359635593043741565, −27.08788275201073927866129340006, −25.20460784476470660795601362024, −24.09683106736943140864086796003, −23.19826051043562525974551567885, −22.42343011226191406651909200181, −20.34368714299720118188396206952, −19.77970036300056360309336359285, −18.463615887409058326352229699936, −16.83776710239368500031202517026, −14.94478900325879985354419080322, −14.097424537131946220508542578858, −12.53851355429131882417369482467, −11.76431731346451045224684145901, −10.404310862759040133167272216772, −7.7520808488237970527744751238, −6.81438443460058307245429856469, −4.8533363327096132208105592679, −3.17326300015674865775396292630, −1.13876102394738274083525472572,
3.18257907924013307436748018584, 4.52985905401191320022209703182, 5.69035964075830544218634997858, 7.76067688800129770689720361133, 9.14308230386593759811049475898, 11.489502316175733790026973955713, 11.91847849506780059768347309850, 14.08959363763419881733290868210, 15.13210195157242102212897073672, 16.06307316678443873114637211200, 17.08245772728451635125580454426, 19.31752266567367975739819342977, 20.74913312661126870608916776904, 21.693754131246465465193212890446, 22.68472416017094808108271005853, 23.98356825726259237943844789240, 25.00626481729132838543650005089, 26.59702503731744895018753914068, 27.47015190747235562510343845671, 28.81753345635915390353954695474, 30.51278555348308758786832462141, 31.77024386347701548440781716266, 32.01769481757096341534737153695, 33.580995004895736549319876157663, 34.43905359629817189293051961685