| L(s) = 1 | − 2·2-s − 4-s + 3·5-s + 4·7-s + 8·8-s − 6·10-s − 3·11-s − 2·13-s − 8·14-s − 7·16-s + 4·17-s + 19-s − 3·20-s + 6·22-s + 5·25-s + 4·26-s − 4·28-s + 7·29-s − 3·31-s − 14·32-s − 8·34-s + 12·35-s + 4·37-s − 2·38-s + 24·40-s + 3·41-s + 7·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.34·5-s + 1.51·7-s + 2.82·8-s − 1.89·10-s − 0.904·11-s − 0.554·13-s − 2.13·14-s − 7/4·16-s + 0.970·17-s + 0.229·19-s − 0.670·20-s + 1.27·22-s + 25-s + 0.784·26-s − 0.755·28-s + 1.29·29-s − 0.538·31-s − 2.47·32-s − 1.37·34-s + 2.02·35-s + 0.657·37-s − 0.324·38-s + 3.79·40-s + 0.468·41-s + 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174917283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174917283\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.19498515193657594743587468463, −9.939165597226645645218230901820, −9.561407804816850601755642414400, −9.343161768955426914645143453045, −8.746883799490829594903641028401, −8.300126757796985391135939740893, −8.044422464239794806373274107640, −7.85170734237166521008911248876, −7.10504011516982068271075081864, −6.94443125250656373439691471542, −5.79822073224444378540588248686, −5.58625417441522171535448808730, −4.99660545715066328729164023005, −4.90940484260439897130647317915, −4.23956269856436241918243244399, −3.62128743612012108477951849056, −2.43760343044770223921198018033, −2.18038063981077940198230753869, −1.23821931987795033679240901262, −0.835655756923601098163415042913,
0.835655756923601098163415042913, 1.23821931987795033679240901262, 2.18038063981077940198230753869, 2.43760343044770223921198018033, 3.62128743612012108477951849056, 4.23956269856436241918243244399, 4.90940484260439897130647317915, 4.99660545715066328729164023005, 5.58625417441522171535448808730, 5.79822073224444378540588248686, 6.94443125250656373439691471542, 7.10504011516982068271075081864, 7.85170734237166521008911248876, 8.044422464239794806373274107640, 8.300126757796985391135939740893, 8.746883799490829594903641028401, 9.343161768955426914645143453045, 9.561407804816850601755642414400, 9.939165597226645645218230901820, 10.19498515193657594743587468463
