| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s + 4·8-s − 4·10-s + 2·11-s + 8·13-s + 8·16-s − 10·17-s + 2·19-s − 4·20-s + 4·22-s + 6·23-s + 3·25-s + 16·26-s − 2·29-s + 6·31-s + 8·32-s − 20·34-s + 4·37-s + 4·38-s − 8·40-s − 2·41-s − 4·43-s + 4·44-s + 12·46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 0.603·11-s + 2.21·13-s + 2·16-s − 2.42·17-s + 0.458·19-s − 0.894·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s + 3.13·26-s − 0.371·29-s + 1.07·31-s + 1.41·32-s − 3.42·34-s + 0.657·37-s + 0.648·38-s − 1.26·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.76·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.091532824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.091532824\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.942765442895249633243179848159, −8.856626119033657402942021422640, −8.551673257961172393688433083216, −7.934143389558448207026804335539, −7.73787208501959780982677005941, −7.17006836423974039603091920964, −6.69304689638824678908000491965, −6.54098959762751938422260283484, −6.11815815165238951394572438151, −5.71365542842930466343236736977, −4.92676084689245965889564072181, −4.84393372257600961888642322455, −4.28490138845435264442344354248, −4.15585111569565795210609654941, −3.49182043444019856818942444692, −3.35453262065369419049874150601, −2.73384196480876704249664549308, −1.89817116903868295491057521179, −1.42328622967406263087918345191, −0.72700534850424245923582315910,
0.72700534850424245923582315910, 1.42328622967406263087918345191, 1.89817116903868295491057521179, 2.73384196480876704249664549308, 3.35453262065369419049874150601, 3.49182043444019856818942444692, 4.15585111569565795210609654941, 4.28490138845435264442344354248, 4.84393372257600961888642322455, 4.92676084689245965889564072181, 5.71365542842930466343236736977, 6.11815815165238951394572438151, 6.54098959762751938422260283484, 6.69304689638824678908000491965, 7.17006836423974039603091920964, 7.73787208501959780982677005941, 7.934143389558448207026804335539, 8.551673257961172393688433083216, 8.856626119033657402942021422640, 8.942765442895249633243179848159
