| L(s) = 1 | + 6·2-s + 11·4-s − 36·8-s + 66·11-s − 267·16-s + 252·17-s + 396·22-s − 142·25-s − 48·29-s − 140·31-s − 558·32-s + 1.51e3·34-s + 364·37-s + 588·41-s + 726·44-s + 398·49-s − 852·50-s − 288·58-s − 840·62-s + 895·64-s − 1.76e3·67-s + 2.77e3·68-s + 2.18e3·74-s + 3.52e3·82-s + 2.43e3·83-s − 2.37e3·88-s − 392·97-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 11/8·4-s − 1.59·8-s + 1.80·11-s − 4.17·16-s + 3.59·17-s + 3.83·22-s − 1.13·25-s − 0.307·29-s − 0.811·31-s − 3.08·32-s + 7.62·34-s + 1.61·37-s + 2.23·41-s + 2.48·44-s + 1.16·49-s − 2.40·50-s − 0.652·58-s − 1.72·62-s + 1.74·64-s − 3.20·67-s + 4.94·68-s + 3.43·74-s + 4.75·82-s + 3.22·83-s − 2.87·88-s − 0.410·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.260465432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.260465432\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 6 p T + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - 3 T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 398 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3512 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5780 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9884 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158996 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 195788 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 276122 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 145766 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 347240 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 880 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 601580 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 746282 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 389846 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1218 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 944512 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.99933741374808131924415334162, −13.24275859997159811606916457759, −12.52276122825959233547411034542, −12.34009633039195019149418368827, −11.81906568912487640316304289130, −11.62531572229395898878488262486, −10.64064200960985064678768025605, −9.655604908610199739306963188426, −9.457393683404052959799211672698, −8.960917921763530838394495914368, −7.88795754273154221829542236793, −7.44749897744490840866445917875, −6.32921320818130283187633644727, −5.75453061583741402606114379149, −5.68665902939013417695247495104, −4.67092600195408856889309482277, −3.84371975304771202768784668169, −3.71157769955466952845691738512, −2.80272473758230927183061136704, −1.06699964224165067041201041074,
1.06699964224165067041201041074, 2.80272473758230927183061136704, 3.71157769955466952845691738512, 3.84371975304771202768784668169, 4.67092600195408856889309482277, 5.68665902939013417695247495104, 5.75453061583741402606114379149, 6.32921320818130283187633644727, 7.44749897744490840866445917875, 7.88795754273154221829542236793, 8.960917921763530838394495914368, 9.457393683404052959799211672698, 9.655604908610199739306963188426, 10.64064200960985064678768025605, 11.62531572229395898878488262486, 11.81906568912487640316304289130, 12.34009633039195019149418368827, 12.52276122825959233547411034542, 13.24275859997159811606916457759, 13.99933741374808131924415334162
