| L(s) = 1 | + 3·4-s − 2·5-s − 4·7-s + 4·11-s + 5·16-s − 2·17-s − 6·20-s − 8·23-s + 2·25-s − 12·28-s + 2·29-s − 8·31-s + 8·35-s − 10·37-s + 2·41-s + 12·44-s + 24·47-s + 8·49-s − 8·55-s + 14·61-s + 3·64-s − 8·67-s − 6·68-s + 20·71-s − 18·73-s − 16·77-s − 24·79-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.20·11-s + 5/4·16-s − 0.485·17-s − 1.34·20-s − 1.66·23-s + 2/5·25-s − 2.26·28-s + 0.371·29-s − 1.43·31-s + 1.35·35-s − 1.64·37-s + 0.312·41-s + 1.80·44-s + 3.50·47-s + 8/7·49-s − 1.07·55-s + 1.79·61-s + 3/8·64-s − 0.977·67-s − 0.727·68-s + 2.37·71-s − 2.10·73-s − 1.82·77-s − 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.687574873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687574873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−10.47774123055211622854236227448, −10.18249845401120163114463512028, −10.06943761811139929736168732774, −9.154471781201814124259538353373, −8.816025749368377569955052572825, −8.743076767565240467406916736167, −7.74486748862615377942964373168, −7.31972724866868153773882166583, −7.27263497824625334841740644712, −6.55739764432784642409077044983, −6.43653788762428107425862491696, −5.77719330677020759978343995383, −5.57828466893917670718242277514, −4.36662203696772474795275079492, −4.09600132524729771615909258650, −3.40693335099078153253249665907, −3.25545580521991274749613444494, −2.25387473243296001443820157835, −1.92385001185675059471069882998, −0.64327460879704836069112371110,
0.64327460879704836069112371110, 1.92385001185675059471069882998, 2.25387473243296001443820157835, 3.25545580521991274749613444494, 3.40693335099078153253249665907, 4.09600132524729771615909258650, 4.36662203696772474795275079492, 5.57828466893917670718242277514, 5.77719330677020759978343995383, 6.43653788762428107425862491696, 6.55739764432784642409077044983, 7.27263497824625334841740644712, 7.31972724866868153773882166583, 7.74486748862615377942964373168, 8.743076767565240467406916736167, 8.816025749368377569955052572825, 9.154471781201814124259538353373, 10.06943761811139929736168732774, 10.18249845401120163114463512028, 10.47774123055211622854236227448
