L(s) = 1 | + (−0.874 − 0.485i)2-s + (−0.874 + 0.485i)3-s + (0.528 + 0.848i)4-s + (−0.612 + 0.790i)5-s + 6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (0.528 − 0.848i)9-s + (0.918 − 0.394i)10-s + (0.250 − 0.968i)11-s + (−0.874 − 0.485i)12-s + (0.528 − 0.848i)13-s + (−0.954 − 0.299i)14-s + (0.151 − 0.988i)15-s + (−0.440 + 0.897i)16-s + (−0.918 + 0.394i)17-s + ⋯ |
L(s) = 1 | + (−0.874 − 0.485i)2-s + (−0.874 + 0.485i)3-s + (0.528 + 0.848i)4-s + (−0.612 + 0.790i)5-s + 6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (0.528 − 0.848i)9-s + (0.918 − 0.394i)10-s + (0.250 − 0.968i)11-s + (−0.874 − 0.485i)12-s + (0.528 − 0.848i)13-s + (−0.954 − 0.299i)14-s + (0.151 − 0.988i)15-s + (−0.440 + 0.897i)16-s + (−0.918 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2046723985 - 0.3466325780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2046723985 - 0.3466325780i\) |
\(L(1)\) |
\(\approx\) |
\(0.5116657533 - 0.04149772258i\) |
\(L(1)\) |
\(\approx\) |
\(0.5116657533 - 0.04149772258i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.874 - 0.485i)T \) |
| 3 | \( 1 + (-0.874 + 0.485i)T \) |
| 5 | \( 1 + (-0.612 + 0.790i)T \) |
| 7 | \( 1 + (0.979 - 0.201i)T \) |
| 11 | \( 1 + (0.250 - 0.968i)T \) |
| 13 | \( 1 + (0.528 - 0.848i)T \) |
| 17 | \( 1 + (-0.918 + 0.394i)T \) |
| 19 | \( 1 + (0.612 + 0.790i)T \) |
| 23 | \( 1 + (0.0506 + 0.998i)T \) |
| 29 | \( 1 + (0.954 - 0.299i)T \) |
| 31 | \( 1 + (-0.918 - 0.394i)T \) |
| 37 | \( 1 + (-0.979 - 0.201i)T \) |
| 41 | \( 1 + (-0.347 - 0.937i)T \) |
| 43 | \( 1 + (-0.979 + 0.201i)T \) |
| 47 | \( 1 + (-0.954 + 0.299i)T \) |
| 53 | \( 1 + (0.979 - 0.201i)T \) |
| 59 | \( 1 + (-0.979 + 0.201i)T \) |
| 61 | \( 1 + (0.612 + 0.790i)T \) |
| 67 | \( 1 + (0.347 - 0.937i)T \) |
| 71 | \( 1 + (-0.820 - 0.571i)T \) |
| 73 | \( 1 + (-0.994 + 0.101i)T \) |
| 79 | \( 1 + (0.347 + 0.937i)T \) |
| 83 | \( 1 + (0.151 + 0.988i)T \) |
| 89 | \( 1 + (0.979 + 0.201i)T \) |
| 97 | \( 1 + (-0.820 - 0.571i)T \) |
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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
−24.91728457361716271460659454843, −24.5555206851939102331902650273, −23.677422032085342289032400380995, −23.140632141417388413097173118541, −21.77287755467470141022246408985, −20.49059662791545141060436587734, −19.86643644550365661023370545188, −18.69216818699989791368807212646, −17.94828767013553380245424169059, −17.307710874728367389359748757569, −16.34158327325673020422535848005, −15.69118556295865407180856701331, −14.58242825595768420982509488500, −13.31171990373980851248097272959, −11.92694419294261534897197818885, −11.54571261295658316444417094536, −10.51288274658760456621280155002, −9.10905388618314528073199230615, −8.3653038204334585551750058595, −7.25553354197895079273546580539, −6.59130278451441297062017511436, −5.05986811459664399215584469297, −4.63686089407130861783071449980, −1.98472603461892463782991529564, −1.12734881054134220730716777057,
0.21318905946978977600880289230, 1.48414436069342611082357910697, 3.2793335881471338032970407756, 4.02766286874136339503931088002, 5.6142496912890002354166007107, 6.7659590169998918244020836654, 7.83387482124146718279186248597, 8.71186252496408732480943872986, 10.13208795478577129494919754981, 10.86846303582083467140284061509, 11.37618108496414089048773413631, 12.14786309969876219251498872857, 13.62022588528107601449543330497, 15.05535706189885614401969235345, 15.78102061114064505598858712374, 16.69884308996306535094976108178, 17.776264150422366918194290405943, 18.13722337714387618145969513426, 19.22065499275366743159042383933, 20.21890762596374248188667486488, 21.16930555755772413337569911338, 21.9194623321877123227437328693, 22.7507635904667156503856230859, 23.80412978916052634119469412182, 24.706727655539342952169567901185