| L(s) = 1 | + 0.618i·2-s + 1.23i·3-s + 1.61·4-s − 0.763·6-s + 4.23i·7-s + 2.23i·8-s + 1.47·9-s + 2·11-s + 2.00i·12-s + 1.23i·13-s − 2.61·14-s + 1.85·16-s − 5.23i·17-s + 0.909i·18-s − 2.23·19-s + ⋯ |
| L(s) = 1 | + 0.437i·2-s + 0.713i·3-s + 0.809·4-s − 0.311·6-s + 1.60i·7-s + 0.790i·8-s + 0.490·9-s + 0.603·11-s + 0.577i·12-s + 0.342i·13-s − 0.699·14-s + 0.463·16-s − 1.26i·17-s + 0.214i·18-s − 0.512·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05338 + 1.70441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05338 + 1.70441i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618iT - 2T^{2} \) |
| 3 | \( 1 - 1.23iT - 3T^{2} \) |
| 7 | \( 1 - 4.23iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 5.23iT - 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.70iT - 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 3.23iT - 43T^{2} \) |
| 47 | \( 1 - 6.47iT - 47T^{2} \) |
| 53 | \( 1 + 1.52iT - 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 0.472iT - 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 2.94iT - 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 1.94iT - 97T^{2} \) |
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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
−10.67642082135070259998986765159, −9.479445446389611621460684200274, −9.059948352083836517861650829432, −8.039695517886971027097047983794, −6.96400736414583977095726956002, −6.22142516575542009858670129554, −5.32298073444073179859727169069, −4.39586386883283072388372367154, −2.95097733758648596568312654931, −1.99419830754532092397817370743,
1.09490794912294377570184876855, 1.89156114847696632872255296589, 3.56655761800560413284489452849, 4.14003559294332340495888200381, 5.88989678311578975235857402178, 6.72614078004165344112620265171, 7.41534415278134864976179208463, 7.918711522066346171742924589414, 9.445673154040015306846335322221, 10.27098456016237301317007727803
