L(s) = 1 | + (−0.371 + 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (−0.909 + 0.415i)5-s + (0.371 + 0.928i)6-s + (−0.814 − 0.580i)7-s + (0.909 − 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (0.814 − 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (−0.371 + 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (0.909 + 0.415i)18-s + ⋯ |
L(s) = 1 | + (−0.371 + 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (−0.909 + 0.415i)5-s + (0.371 + 0.928i)6-s + (−0.814 − 0.580i)7-s + (0.909 − 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (0.814 − 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (−0.371 + 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (0.909 + 0.415i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.316363182 - 0.6517383274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316363182 - 0.6517383274i\) |
\(L(1)\) |
\(\approx\) |
\(0.8934599547 + 0.02107686732i\) |
\(L(1)\) |
\(\approx\) |
\(0.8934599547 + 0.02107686732i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (-0.371 + 0.928i)T \) |
| 3 | \( 1 + (0.723 - 0.690i)T \) |
| 5 | \( 1 + (-0.909 + 0.415i)T \) |
| 7 | \( 1 + (-0.814 - 0.580i)T \) |
| 11 | \( 1 + (0.814 - 0.580i)T \) |
| 17 | \( 1 + (-0.928 + 0.371i)T \) |
| 19 | \( 1 + (0.998 + 0.0475i)T \) |
| 23 | \( 1 + (0.888 - 0.458i)T \) |
| 29 | \( 1 + (-0.995 - 0.0950i)T \) |
| 31 | \( 1 + (0.540 + 0.841i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.971 + 0.235i)T \) |
| 43 | \( 1 + (0.995 - 0.0950i)T \) |
| 47 | \( 1 + (0.281 + 0.959i)T \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.690 - 0.723i)T \) |
| 61 | \( 1 + (-0.327 + 0.945i)T \) |
| 67 | \( 1 + (0.690 + 0.723i)T \) |
| 71 | \( 1 + (-0.0950 - 0.995i)T \) |
| 73 | \( 1 + (-0.540 - 0.841i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.989 - 0.142i)T \) |
| 97 | \( 1 + (0.814 + 0.580i)T \) |
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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
−20.96401141484198908761959395031, −20.20595959863124117942935484420, −19.83549701024519484253141586448, −19.157363282580918164833614451342, −18.49618076558285758719562895227, −17.31233914523036117594642401811, −16.52074879852283336368979265464, −15.766164901243373518616874591361, −15.13457781206658328972025512985, −14.12089009257636605990250282160, −13.13389915677617630585451925489, −12.59041943068371030615720715181, −11.54385267351914471326750101785, −11.12494620712962216681085197658, −9.8270536291574989550691785654, −9.2506130473997365569802918279, −8.90562895603580839983758964104, −7.79859390590205881027072520632, −7.0989720958879356473188994246, −5.42143769028251744837636148416, −4.42310467576516586252294582432, −3.834406960609225957373361851856, −3.00627791206949134453639689759, −2.16373140509565097816650796842, −0.81910157008866267138333476680,
0.45509726813511919427278871880, 1.209255617736235182672196275902, 2.856736438017436448463202837725, 3.70311988188837658854416593196, 4.46809770338462867461388439327, 6.08242968240169949732012001074, 6.649247577998757692145601652289, 7.327595469872840101725565257327, 7.97149729271812518328758701211, 8.936520069802644725508531298937, 9.43109654727970434297414178398, 10.61625355890945896833809297807, 11.47425738650818568829327381796, 12.65643043854408313520073081687, 13.32329565047920126755309023338, 14.180415547079793734140794523394, 14.69109733451275630910636488112, 15.61355450376922482067194679278, 16.191844108242341775224715632, 17.10665126535080466402872440764, 17.929619085052140812978502325595, 18.82198363029680085497624318590, 19.33909554613011157682505018989, 19.7658104204041053469581620909, 20.60905080865909272941777282366