L(s) = 1 | + (−0.550 + 0.834i)2-s + (−0.0448 − 0.998i)3-s + (−0.393 − 0.919i)4-s + (−0.0448 + 0.998i)5-s + (0.858 + 0.512i)6-s + (−0.550 − 0.834i)7-s + (0.983 + 0.178i)8-s + (−0.995 + 0.0896i)9-s + (−0.809 − 0.587i)10-s + (−0.963 − 0.266i)11-s + (−0.900 + 0.433i)12-s + (−0.691 + 0.722i)13-s + 14-s + 15-s + (−0.691 + 0.722i)16-s + (0.134 + 0.990i)17-s + ⋯ |
L(s) = 1 | + (−0.550 + 0.834i)2-s + (−0.0448 − 0.998i)3-s + (−0.393 − 0.919i)4-s + (−0.0448 + 0.998i)5-s + (0.858 + 0.512i)6-s + (−0.550 − 0.834i)7-s + (0.983 + 0.178i)8-s + (−0.995 + 0.0896i)9-s + (−0.809 − 0.587i)10-s + (−0.963 − 0.266i)11-s + (−0.900 + 0.433i)12-s + (−0.691 + 0.722i)13-s + 14-s + 15-s + (−0.691 + 0.722i)16-s + (0.134 + 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01261795188 + 0.1415482457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01261795188 + 0.1415482457i\) |
\(L(1)\) |
\(\approx\) |
\(0.4850393114 + 0.09994866231i\) |
\(L(1)\) |
\(\approx\) |
\(0.4850393114 + 0.09994866231i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 211 | \( 1 \) |
good | 2 | \( 1 + (-0.550 + 0.834i)T \) |
| 3 | \( 1 + (-0.0448 - 0.998i)T \) |
| 5 | \( 1 + (-0.0448 + 0.998i)T \) |
| 7 | \( 1 + (-0.550 - 0.834i)T \) |
| 11 | \( 1 + (-0.963 - 0.266i)T \) |
| 13 | \( 1 + (-0.691 + 0.722i)T \) |
| 17 | \( 1 + (0.134 + 0.990i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.691 + 0.722i)T \) |
| 31 | \( 1 + (-0.900 + 0.433i)T \) |
| 37 | \( 1 + (-0.963 + 0.266i)T \) |
| 41 | \( 1 + (0.473 - 0.880i)T \) |
| 43 | \( 1 + (-0.900 + 0.433i)T \) |
| 47 | \( 1 + (-0.995 - 0.0896i)T \) |
| 53 | \( 1 + (-0.393 + 0.919i)T \) |
| 59 | \( 1 + (0.473 - 0.880i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + (0.983 - 0.178i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.936 - 0.351i)T \) |
| 97 | \( 1 + (0.753 - 0.657i)T \) |
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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
−26.52685436346853399557131395047, −25.49603144912473311433479142310, −24.71069116741484738909392245338, −22.92346228470589966615154256225, −22.362537062560115093688824699302, −21.18854011466987157903140150427, −20.659463340523034990683371420923, −19.90587010481613631473282114565, −18.72008027954535797218715775797, −17.71582887566059977888045630180, −16.539383103002214707751774417660, −16.07001191231005694709861840931, −14.884082728484867064552400424673, −13.25226529166705031256431457493, −12.4163867364714127882415558251, −11.57822510866517466585297221105, −10.203856778172490712972309851361, −9.62099862210425758148009869863, −8.70508690935860870715565923386, −7.74524800917025264660918213513, −5.52763750598127800149007562319, −4.76543239207963856651891091307, −3.3763951670632806554949351886, −2.32167158056225664666490611226, −0.1210565277183089529994733963,
1.78682839795819472178281660877, 3.33291019641558016854240951076, 5.24351301838699273746500971779, 6.46545987231845628736207214124, 7.16601350044247588692203148541, 7.80098151188429031857875483209, 9.22281777553877077030486233398, 10.44691021295718772158171942866, 11.22674745863479490782053793653, 12.91566489490706650680639010633, 13.76589145717811548253040474520, 14.55988434733765903689065874022, 15.679152916506957282245390463479, 16.86844657973229483386138149274, 17.6190251922468933215868838550, 18.55844570450544450541234708900, 19.283292098742536007966805448783, 19.92115819425197091555186241146, 21.805192197201239540984529679935, 22.86046148272596314818857390590, 23.697150227322997363510433074308, 24.093312220733002776243596139394, 25.538266972605372900823153572065, 26.139765588411946002151321847313, 26.63216238850156697583944830565