L(s) = 1 | + (−0.528 − 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (−0.683 − 0.729i)11-s + (0.634 + 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.793 − 0.608i)17-s + (−0.412 − 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.812 − 0.582i)37-s + ⋯ |
L(s) = 1 | + (−0.528 − 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (−0.683 − 0.729i)11-s + (0.634 + 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.793 − 0.608i)17-s + (−0.412 − 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.812 − 0.582i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.03873155434 - 0.9018138596i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.03873155434 - 0.9018138596i\) |
\(L(1)\) |
\(\approx\) |
\(0.6914065483 - 0.4783886829i\) |
\(L(1)\) |
\(\approx\) |
\(0.6914065483 - 0.4783886829i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.528 - 0.849i)T \) |
| 5 | \( 1 + (0.162 - 0.986i)T \) |
| 11 | \( 1 + (-0.683 - 0.729i)T \) |
| 13 | \( 1 + (0.634 + 0.773i)T \) |
| 17 | \( 1 + (0.793 - 0.608i)T \) |
| 19 | \( 1 + (-0.412 - 0.910i)T \) |
| 23 | \( 1 + (-0.321 - 0.946i)T \) |
| 29 | \( 1 + (0.956 - 0.290i)T \) |
| 31 | \( 1 + (0.258 + 0.965i)T \) |
| 37 | \( 1 + (0.812 - 0.582i)T \) |
| 41 | \( 1 + (-0.195 + 0.980i)T \) |
| 43 | \( 1 + (-0.471 - 0.881i)T \) |
| 47 | \( 1 + (-0.991 + 0.130i)T \) |
| 53 | \( 1 + (-0.729 + 0.683i)T \) |
| 59 | \( 1 + (-0.352 - 0.935i)T \) |
| 61 | \( 1 + (-0.0327 - 0.999i)T \) |
| 67 | \( 1 + (0.849 - 0.528i)T \) |
| 71 | \( 1 + (-0.555 + 0.831i)T \) |
| 73 | \( 1 + (-0.0654 - 0.997i)T \) |
| 79 | \( 1 + (0.608 - 0.793i)T \) |
| 83 | \( 1 + (0.0980 - 0.995i)T \) |
| 89 | \( 1 + (-0.659 - 0.751i)T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−20.88706569135898765535224824000, −19.93037075528213460434299125140, −19.02463060330921823763402332476, −18.12327638399310644409809616149, −17.783645625829397952389844977530, −16.9097918759910577738675261010, −16.09945619009614462399632002942, −15.25298695565655981588364244543, −14.971013159250298427480831548966, −14.08911459831823542735722968032, −13.115171640583079396757498084123, −12.26863840862452976226983407960, −11.436517618886568148777211646427, −10.6996396746462371891959274500, −10.03643367627166423739484904611, −9.758579819607820403836864063747, −8.32884988998536056920311602058, −7.7279097529524735579614333842, −6.58877736920226131108513805703, −5.903369428952255809203526646093, −5.28002775552791123721479308161, −4.14801354267753990254160559046, −3.44591191588213969879001603951, −2.63950296155042080510277146528, −1.37078290296317948655600187061,
0.37441759925199535471398571593, 1.21929781201898156464104886805, 2.19668191834480232595085610757, 3.18060126476387559226992493424, 4.613910034400541862352315825212, 5.05613464308642147204041851922, 6.096690567594461515040818768158, 6.57503695139381212995854683592, 7.7425352401345053230552585341, 8.35701690982250886884882987776, 9.01874222329875584217040279264, 10.10032768079401646071144698230, 11.00934194252906706271302995927, 11.68119767493718107340584707849, 12.42698880713513822878143506760, 13.076032968061576487892389724277, 13.74334484831438709088483502727, 14.30369951896141503575291392793, 15.80029940819980433341733928896, 16.23115382466768573009454605927, 16.856447540999754144762023716372, 17.64765417494871703542450304296, 18.38981528599579021713385283707, 18.95938400802350256232009307056, 19.76922273766015802675357670552