L(s) = 1 | + 2.09·2-s + (−1.16 − 1.28i)3-s + 2.38·4-s + 1.47·5-s + (−2.44 − 2.68i)6-s − 2.11i·7-s + 0.798·8-s + (−0.281 + 2.98i)9-s + 3.09·10-s + 1.98·11-s + (−2.77 − 3.05i)12-s + 0.757i·13-s − 4.43i·14-s + (−1.72 − 1.89i)15-s − 3.09·16-s + 4.64i·17-s + ⋯ |
L(s) = 1 | + 1.48·2-s + (−0.673 − 0.739i)3-s + 1.19·4-s + 0.660·5-s + (−0.996 − 1.09i)6-s − 0.800i·7-s + 0.282·8-s + (−0.0936 + 0.995i)9-s + 0.977·10-s + 0.598·11-s + (−0.801 − 0.880i)12-s + 0.210i·13-s − 1.18i·14-s + (−0.444 − 0.488i)15-s − 0.772·16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01567 - 0.718661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01567 - 0.718661i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.16 + 1.28i)T \) |
| 67 | \( 1 + (-8.09 + 1.20i)T \) |
good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 + 2.11iT - 7T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 - 0.757iT - 13T^{2} \) |
| 17 | \( 1 - 4.64iT - 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 - 3.56iT - 23T^{2} \) |
| 29 | \( 1 - 2.91iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 5.79iT - 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 14.2iT - 59T^{2} \) |
| 61 | \( 1 + 10.7iT - 61T^{2} \) |
| 71 | \( 1 - 4.38iT - 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 17.6iT - 79T^{2} \) |
| 83 | \( 1 - 5.31iT - 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 - 7.55iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.69066010601471145115428252805, −11.61595874621975835218799390799, −10.89051699693194430810265331011, −9.604256127849149452818850482725, −7.914090106958505160733689215372, −6.60673778769647911376793289088, −6.08828543004255678773501930040, −4.92902559242463142325646638690, −3.74061805522464458864914007246, −1.86415770901429275964449738495,
2.67410437813553966203515815590, 4.08587404700328560041364707338, 5.15767458302728102532633062293, 5.88026187296419329668294204013, 6.75655506146804001859374420058, 8.864465736249618275842881590495, 9.713926736591871076547243532186, 10.93931917492429819540774232602, 11.92049648675857254088065020220, 12.39380805949765150193302313688