L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)6-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.5 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)6-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.173 − 0.984i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.5 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5003387464 + 0.02942905175i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5003387464 + 0.02942905175i\) |
\(L(1)\) |
\(\approx\) |
\(0.4936292289 + 0.3683732114i\) |
\(L(1)\) |
\(\approx\) |
\(0.4936292289 + 0.3683732114i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.939 + 0.342i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.766 + 0.642i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.939 + 0.342i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.37300139312716746386292214889, −27.79336861365713775299852140742, −26.52700192908394091256389553066, −25.43643471158185140924433496850, −24.1267403526926957204739982353, −23.10992081849907684559908988381, −22.68703018093370035601459964895, −20.98088882038132840769714957549, −20.08687085522744149469592533991, −19.37890857371400177285204036267, −18.29769747598190393284064995504, −17.63831907956873593641345948377, −16.25981960948772952201959774077, −14.81265508779835013990904886523, −13.17815008799091868388996581205, −12.85529603334960354692885741057, −11.55689249853038978695172973619, −10.97997507188248086102739952587, −9.237231083887792879414322403754, −8.14440881557700525060436755116, −7.236379104720406186810604838302, −5.36135341489824125634882606488, −3.992664013680190017302944862327, −2.51766238355344573741833859822, −1.0797812848553048723676537409,
0.267051417697106983176985368130, 3.41596374164072654880342395127, 4.38862802201736263752239448161, 5.66532805696909827577184165487, 6.85420988829905768287828860357, 8.32657307061190776632001984478, 8.9915399903450021616254815147, 10.58423154313320192865471970410, 11.26632992041949095775107869748, 13.08429275658814766655816535042, 14.39501053027624780873889291147, 15.281864591934868383756321726771, 16.06116118568160284698188558910, 16.73900168079293295820646356494, 18.153087461758786213301157117653, 19.05861429935183729820315214307, 20.32667758061850800787521047840, 21.67272141279305863385352182183, 22.55186530153216143352617661849, 23.46039770067246517911835176939, 24.219499940707891221730983720, 25.78066241783274397999406136580, 26.45109537827143581480324297077, 27.172574546580620784414541027287, 27.99862309474392101880272254116