L(s) = 1 | + (−0.0771 − 0.997i)2-s + (−0.843 − 0.536i)3-s + (−0.988 + 0.153i)4-s + (−0.967 − 0.254i)5-s + (−0.469 + 0.882i)6-s + (0.916 − 0.400i)7-s + (0.229 + 0.973i)8-s + (0.423 + 0.905i)9-s + (−0.179 + 0.983i)10-s + (−0.935 + 0.352i)11-s + (0.916 + 0.400i)12-s + (0.423 + 0.905i)13-s + (−0.469 − 0.882i)14-s + (0.679 + 0.733i)15-s + (0.952 − 0.304i)16-s + (0.0257 − 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.0771 − 0.997i)2-s + (−0.843 − 0.536i)3-s + (−0.988 + 0.153i)4-s + (−0.967 − 0.254i)5-s + (−0.469 + 0.882i)6-s + (0.916 − 0.400i)7-s + (0.229 + 0.973i)8-s + (0.423 + 0.905i)9-s + (−0.179 + 0.983i)10-s + (−0.935 + 0.352i)11-s + (0.916 + 0.400i)12-s + (0.423 + 0.905i)13-s + (−0.469 − 0.882i)14-s + (0.679 + 0.733i)15-s + (0.952 − 0.304i)16-s + (0.0257 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5430988264 - 0.4451988868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5430988264 - 0.4451988868i\) |
\(L(1)\) |
\(\approx\) |
\(0.5676873941 - 0.3706608425i\) |
\(L(1)\) |
\(\approx\) |
\(0.5676873941 - 0.3706608425i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (-0.0771 - 0.997i)T \) |
| 3 | \( 1 + (-0.843 - 0.536i)T \) |
| 5 | \( 1 + (-0.967 - 0.254i)T \) |
| 7 | \( 1 + (0.916 - 0.400i)T \) |
| 11 | \( 1 + (-0.935 + 0.352i)T \) |
| 13 | \( 1 + (0.423 + 0.905i)T \) |
| 17 | \( 1 + (0.0257 - 0.999i)T \) |
| 19 | \( 1 + (0.514 + 0.857i)T \) |
| 23 | \( 1 + (0.952 + 0.304i)T \) |
| 29 | \( 1 + (0.815 + 0.579i)T \) |
| 31 | \( 1 + (-0.988 + 0.153i)T \) |
| 37 | \( 1 + (-0.376 + 0.926i)T \) |
| 41 | \( 1 + (-0.935 - 0.352i)T \) |
| 43 | \( 1 + (0.514 - 0.857i)T \) |
| 47 | \( 1 + (0.978 - 0.204i)T \) |
| 53 | \( 1 + (-0.967 - 0.254i)T \) |
| 59 | \( 1 + (0.600 - 0.799i)T \) |
| 61 | \( 1 + (-0.469 - 0.882i)T \) |
| 67 | \( 1 + (0.600 + 0.799i)T \) |
| 71 | \( 1 + (-0.988 - 0.153i)T \) |
| 73 | \( 1 + (0.328 + 0.944i)T \) |
| 79 | \( 1 + (0.679 - 0.733i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.978 - 0.204i)T \) |
| 97 | \( 1 + (0.952 + 0.304i)T \) |
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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
−24.61109930256703962088659887446, −23.798374275407232927207119234692, −23.36237181185519961529760280737, −22.45193594044751475060702111034, −21.63380960658719629245527507774, −20.6982823469778924752223226264, −19.29665636151673879427459066418, −18.256367382312326069046091311710, −17.7930752591304278134787062336, −16.77014454449551151866080768979, −15.73955452827185193758034837791, −15.370306915240993134762527752209, −14.61582630792340457431709183540, −13.16833864459977407741212890148, −12.235781093167631889910121844026, −10.98863556947242050665599150745, −10.54580522764109372170389912045, −9.012088085940340841029990779540, −8.13487003722843051707644658301, −7.31990638263073497311743861677, −6.05827251245317220853544552718, −5.211396185765056730474291052426, −4.43540964535429039212720757961, −3.23908635040294757123526555881, −0.756672213328544554171997380741,
0.888882639272020728239651935666, 1.93401942040415234898805647013, 3.50792696141801756300919989771, 4.76622419626291786400895320792, 5.182906679339164852475831020652, 7.11161424045425413967095450025, 7.82412296767393785285320019766, 8.83314068187456937880279876599, 10.29042818520171248439871552039, 11.09400107686968086745624860758, 11.72750027322085017833979120820, 12.41735534483268716638586453195, 13.44574007497904919034228581642, 14.301142022196880088570539849881, 15.77131910644990347502039332288, 16.7044856418928204025403385613, 17.61565811198081566348423091920, 18.5948760158585069675930795678, 18.8909117142446057657766008714, 20.30478289641353874028094982812, 20.72801897588379151961981238056, 21.82984778003089555484890508774, 22.88119851304349138901053273517, 23.57474374801244674795070784197, 23.88560729652330515473463233410