| L(s) = 1 | − 2·2-s + 4-s + 2·5-s − 4·7-s − 4·10-s + 4·11-s + 8·14-s + 16-s + 4·17-s − 4·19-s + 2·20-s − 8·22-s + 3·25-s − 4·28-s − 12·31-s + 2·32-s − 8·34-s − 8·35-s + 8·38-s − 12·41-s − 8·43-s + 4·44-s − 4·47-s + 6·49-s − 6·50-s + 12·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s − 1.26·10-s + 1.20·11-s + 2.13·14-s + 1/4·16-s + 0.970·17-s − 0.917·19-s + 0.447·20-s − 1.70·22-s + 3/5·25-s − 0.755·28-s − 2.15·31-s + 0.353·32-s − 1.37·34-s − 1.35·35-s + 1.29·38-s − 1.87·41-s − 1.21·43-s + 0.603·44-s − 0.583·47-s + 6/7·49-s − 0.848·50-s + 1.64·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.57217549121920211992053588416, −7.52886712051060729481767135126, −6.84934686728109547250399492504, −6.84213309101825448192302116627, −6.39038992445616430431108062237, −6.03244210365692689303035588180, −5.85772016616454664712203562348, −5.39642316743019549810323972087, −4.84270209104609745983673004590, −4.68327291456368613306287374612, −3.78819964254667588955534554015, −3.58593755683467233224229893063, −3.46191420187300600727087004119, −2.89093165954633791788626066223, −2.19658172720292888016074320731, −1.99112734096043626432814355623, −1.29702326276380121624726396088, −1.00582342729198148068449673699, 0, 0,
1.00582342729198148068449673699, 1.29702326276380121624726396088, 1.99112734096043626432814355623, 2.19658172720292888016074320731, 2.89093165954633791788626066223, 3.46191420187300600727087004119, 3.58593755683467233224229893063, 3.78819964254667588955534554015, 4.68327291456368613306287374612, 4.84270209104609745983673004590, 5.39642316743019549810323972087, 5.85772016616454664712203562348, 6.03244210365692689303035588180, 6.39038992445616430431108062237, 6.84213309101825448192302116627, 6.84934686728109547250399492504, 7.52886712051060729481767135126, 7.57217549121920211992053588416
