L(s) = 1 | + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + 6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.766 − 0.642i)12-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)14-s + (−0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + 6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.766 − 0.642i)12-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)14-s + (−0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.123945752 - 0.2318234829i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123945752 - 0.2318234829i\) |
\(L(1)\) |
\(\approx\) |
\(1.347420239 - 0.2287507207i\) |
\(L(1)\) |
\(\approx\) |
\(1.347420239 - 0.2287507207i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.173 + 0.984i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−35.7703690424167952449320709890, −34.61563161729989997325206011140, −33.0696165797525618153859096930, −31.84538321753655712033758926590, −31.23062148169681445397521129665, −30.16997334636953014283628862830, −28.73575458384905455903997831915, −26.6256749590172552109719102586, −25.89515557935765105374071538156, −24.577349248285784270916030408058, −23.58878149714085156211066988945, −22.66846474768538584699689606269, −20.72885793066942896324800183383, −19.824548167775281700992456078629, −18.31220910999250330441608291573, −16.428492328543676287268762818774, −15.460526412939500344942647906039, −14.045796791726876988710876241740, −12.89178225691047354786806851804, −11.895926931418480777239949652663, −9.17730538607335376462320652680, −7.64113175537467949013766673853, −6.765866126230916827898516123808, −4.4460802596376722671198145886, −2.98263622148487153915444197170,
2.967734476261450887825855095698, 3.78797158344529489509425318575, 5.75985912152735815427950173459, 8.0235563851890551763504913354, 9.82422325404995403762387951091, 10.9811524114810998824101652566, 12.594092042769895040719455731119, 13.90094751931467527280910022890, 15.355396938052683991176612841700, 15.92157181570684824718890933585, 18.88295180225782784345203103884, 19.54182888502293959624360037605, 20.74871436036363558226284951216, 22.03461877886432147840840443950, 22.93252993072868225392483944010, 24.42760485373204856798483069660, 25.93334213200739237931276883996, 27.21396259929731184328147628795, 28.31421453516343753792432814698, 29.8344443490234432809352337284, 30.9933927822704909778004954614, 31.920344180466605436220985914107, 32.56058697222732123543582988990, 34.1260552475515047475748776447, 35.591005891246716537339542284907