L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.396 + 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.957 − 0.286i)5-s + (0.993 − 0.116i)6-s + (−0.686 + 0.727i)7-s + 8-s + (−0.686 − 0.727i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.597 − 0.802i)12-s + (−0.286 − 0.957i)13-s + (0.973 + 0.230i)14-s + (−0.116 + 0.993i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.396 + 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.957 − 0.286i)5-s + (0.993 − 0.116i)6-s + (−0.686 + 0.727i)7-s + 8-s + (−0.686 − 0.727i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.597 − 0.802i)12-s + (−0.286 − 0.957i)13-s + (0.973 + 0.230i)14-s + (−0.116 + 0.993i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5246140196 - 0.8804888448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5246140196 - 0.8804888448i\) |
\(L(1)\) |
\(\approx\) |
\(0.7519490295 - 0.2488865081i\) |
\(L(1)\) |
\(\approx\) |
\(0.7519490295 - 0.2488865081i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.396 + 0.918i)T \) |
| 5 | \( 1 + (0.957 - 0.286i)T \) |
| 7 | \( 1 + (-0.686 + 0.727i)T \) |
| 11 | \( 1 + (0.727 - 0.686i)T \) |
| 13 | \( 1 + (-0.286 - 0.957i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.230 + 0.973i)T \) |
| 31 | \( 1 + (0.998 - 0.0581i)T \) |
| 41 | \( 1 + (0.642 - 0.766i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (0.116 - 0.993i)T \) |
| 53 | \( 1 + (-0.116 - 0.993i)T \) |
| 59 | \( 1 + (-0.993 - 0.116i)T \) |
| 61 | \( 1 + (-0.998 - 0.0581i)T \) |
| 67 | \( 1 + (0.802 + 0.597i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.973 + 0.230i)T \) |
| 79 | \( 1 + (0.396 + 0.918i)T \) |
| 83 | \( 1 + (0.0581 - 0.998i)T \) |
| 89 | \( 1 + (0.727 + 0.686i)T \) |
| 97 | \( 1 + (-0.998 - 0.0581i)T \) |
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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
−18.70521124231970928913603517305, −17.73433595358595896109365073028, −17.297420253050543052950293714759, −16.92269007281914723402998926658, −16.30702898795783291872526148871, −15.32032827623673768198602038011, −14.40404103988105629719603950161, −13.92647349527248735028972853107, −13.54296274177661576170321114106, −12.64837323711635451887962002841, −11.93445541778551117831706629060, −10.96071394476124138545454256168, −10.20004623381682494849229810522, −9.59563526258964770537963270064, −9.13497641128436743136087218367, −7.8634003622779498929287044118, −7.46115409765011405424026464658, −6.68889206003681599783108764143, −6.20732569674483664066116910256, −5.73141627146247105393789354456, −4.69601963640666579034749421999, −3.83048656632654721035211365013, −2.544126544663189741210994659941, −1.478226477016272015540009115253, −1.2386114602944736327995026207,
0.41282298394635575611150770485, 1.17574699545133233545576766334, 2.59468518014332595204242849050, 2.919531743233783577162593786054, 3.70982969162392299521946531523, 4.83339677392060824021056948104, 5.31459037971985025568749783301, 6.103798563176520759326679031315, 6.918327313698545037527705527256, 8.19767791016457364206168698796, 8.91109332967981202060212098391, 9.40135952702518128985360243717, 9.85822987729981321489445829789, 10.55494006164738460441119949924, 11.23427824322588659506382197373, 12.1126504517060037040972245622, 12.43564338399995195164078878988, 13.37928049805140605648621868946, 14.04975935686775692193831292029, 14.79827333927995174799362149427, 15.89800838349428358891665976484, 16.32314669516646957957384157285, 16.92086269586709423934760644531, 17.66264512568812276900663048979, 18.14490662683096883268867690993