L(s) = 1 | + (−0.749 + 0.662i)2-s + (0.123 − 0.992i)4-s + (−0.991 + 0.132i)5-s + (−0.345 − 0.938i)7-s + (0.564 + 0.825i)8-s + (0.654 − 0.755i)10-s + (0.683 − 0.730i)13-s + (0.879 + 0.475i)14-s + (−0.969 − 0.244i)16-s + (0.362 + 0.931i)17-s + (−0.774 − 0.633i)19-s + (0.00951 + 0.999i)20-s + (−0.786 − 0.618i)23-s + (0.964 − 0.263i)25-s + (−0.0285 + 0.999i)26-s + ⋯ |
L(s) = 1 | + (−0.749 + 0.662i)2-s + (0.123 − 0.992i)4-s + (−0.991 + 0.132i)5-s + (−0.345 − 0.938i)7-s + (0.564 + 0.825i)8-s + (0.654 − 0.755i)10-s + (0.683 − 0.730i)13-s + (0.879 + 0.475i)14-s + (−0.969 − 0.244i)16-s + (0.362 + 0.931i)17-s + (−0.774 − 0.633i)19-s + (0.00951 + 0.999i)20-s + (−0.786 − 0.618i)23-s + (0.964 − 0.263i)25-s + (−0.0285 + 0.999i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7655091197 - 0.4157090275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7655091197 - 0.4157090275i\) |
\(L(1)\) |
\(\approx\) |
\(0.6252136208 + 0.03228386109i\) |
\(L(1)\) |
\(\approx\) |
\(0.6252136208 + 0.03228386109i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.749 + 0.662i)T \) |
| 5 | \( 1 + (-0.991 + 0.132i)T \) |
| 7 | \( 1 + (-0.345 - 0.938i)T \) |
| 13 | \( 1 + (0.683 - 0.730i)T \) |
| 17 | \( 1 + (0.362 + 0.931i)T \) |
| 19 | \( 1 + (-0.774 - 0.633i)T \) |
| 23 | \( 1 + (-0.786 - 0.618i)T \) |
| 29 | \( 1 + (0.710 - 0.703i)T \) |
| 31 | \( 1 + (0.820 + 0.572i)T \) |
| 37 | \( 1 + (0.516 + 0.856i)T \) |
| 41 | \( 1 + (0.595 - 0.803i)T \) |
| 43 | \( 1 + (-0.723 - 0.690i)T \) |
| 47 | \( 1 + (0.953 + 0.299i)T \) |
| 53 | \( 1 + (0.696 + 0.717i)T \) |
| 59 | \( 1 + (-0.398 + 0.917i)T \) |
| 61 | \( 1 + (0.948 - 0.318i)T \) |
| 67 | \( 1 + (0.580 + 0.814i)T \) |
| 71 | \( 1 + (-0.998 - 0.0570i)T \) |
| 73 | \( 1 + (-0.897 - 0.441i)T \) |
| 79 | \( 1 + (0.761 + 0.647i)T \) |
| 83 | \( 1 + (-0.999 + 0.0380i)T \) |
| 89 | \( 1 + (0.841 + 0.540i)T \) |
| 97 | \( 1 + (-0.991 - 0.132i)T \) |
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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
−21.27759079872131280387565313723, −20.47560932984334513520447725704, −19.666063532526459697371840487264, −19.01733147017675029317819799067, −18.50156673856612041937448201454, −17.75952884397071656250373625291, −16.45600866805221926704140458112, −16.195741217761529143169387999679, −15.39939459658155329919823778293, −14.325969454126285054639830587131, −13.18744219848428714232182505433, −12.38504080232172177743500763125, −11.73597062730348490541830331231, −11.26695584872987736280739365712, −10.13972869638023651112227910773, −9.31849851698840947766411187861, −8.562798167666721314803571438, −7.95448342025903179313937860781, −6.97213708463076165624833600126, −6.01962009529362773919488908307, −4.63843646112461562029065160898, −3.77876475373686932476010470121, −2.94618083735096790629424557445, −1.93939326263835372040320038696, −0.748397057034617996084299314063,
0.38101608073849419392102473381, 1.10612891686000035528689550027, 2.65811915855414752969476370089, 3.89396905456933997270172865314, 4.56842548960955704338896405327, 5.92988236517811071117061920793, 6.62821610129622195654606528050, 7.46161139940513029933276790685, 8.23310773030776579786154560815, 8.73226418641155449094890099729, 10.254649881987649827145640022396, 10.400973113321647113090669205092, 11.35417911280953745635263084523, 12.40265812610225420350893533553, 13.41471343947321890887486978610, 14.22800227374887851228619035940, 15.20128793486298458782251643515, 15.65279151224852046563519252530, 16.476951688536463610464049891827, 17.173209401601719769702399301774, 17.90640548502439297488454896207, 18.914167623473731307507786065505, 19.41767674956185321740943495843, 20.10675240462467548061234650872, 20.75577808023595565625711445040