| L(s) = 1 | + 2.49·2-s + 4.20·4-s − 2.20·5-s + 5.49·8-s − 5.49·10-s + 11-s − 3.28·13-s + 5.26·16-s − 1.49·17-s − 6.91·19-s − 9.26·20-s + 2.49·22-s − 6.49·23-s − 0.140·25-s − 8.18·26-s + 1.64·29-s − 2.35·31-s + 2.14·32-s − 3.71·34-s − 5.55·37-s − 17.2·38-s − 12.1·40-s + 11.2·41-s + 5.26·43-s + 4.20·44-s − 16.1·46-s − 1.49·47-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.10·4-s − 0.985·5-s + 1.94·8-s − 1.73·10-s + 0.301·11-s − 0.911·13-s + 1.31·16-s − 0.361·17-s − 1.58·19-s − 2.07·20-s + 0.531·22-s − 1.35·23-s − 0.0281·25-s − 1.60·26-s + 0.306·29-s − 0.422·31-s + 0.378·32-s − 0.636·34-s − 0.913·37-s − 2.79·38-s − 1.91·40-s + 1.75·41-s + 0.803·43-s + 0.633·44-s − 2.38·46-s − 0.217·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 0.304T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.57T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 1.93T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 - 1.85T + 97T^{2} \) |
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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
−7.58696822247540420046623672411, −7.06486496809310073415735655008, −6.22658376654785748921720957127, −5.70399628029713812386094029891, −4.59401193275718069199032147900, −4.28257365799866261771771669065, −3.65547976724929979972409842463, −2.65129046799518917911940428517, −1.92780555516305087462063248204, 0,
1.92780555516305087462063248204, 2.65129046799518917911940428517, 3.65547976724929979972409842463, 4.28257365799866261771771669065, 4.59401193275718069199032147900, 5.70399628029713812386094029891, 6.22658376654785748921720957127, 7.06486496809310073415735655008, 7.58696822247540420046623672411
