| L(s) = 1 | − 4-s + 5-s − 2·9-s − 3·16-s − 3·19-s − 20-s + 25-s + 4·31-s + 2·36-s − 12·41-s − 2·45-s + 2·49-s + 24·59-s + 4·61-s + 7·64-s + 12·71-s + 3·76-s − 8·79-s − 3·80-s − 5·81-s − 3·95-s − 100-s + 12·101-s − 8·109-s − 22·121-s − 4·124-s + 125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.447·5-s − 2/3·9-s − 3/4·16-s − 0.688·19-s − 0.223·20-s + 1/5·25-s + 0.718·31-s + 1/3·36-s − 1.87·41-s − 0.298·45-s + 2/7·49-s + 3.12·59-s + 0.512·61-s + 7/8·64-s + 1.42·71-s + 0.344·76-s − 0.900·79-s − 0.335·80-s − 5/9·81-s − 0.307·95-s − 0.0999·100-s + 1.19·101-s − 0.766·109-s − 2·121-s − 0.359·124-s + 0.0894·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6335843323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6335843323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.26957566008800311044646337672, −12.63529765841231317434736409625, −11.75826116113281771370809993152, −11.43511929194894990206894488011, −10.56134276430848544317017291752, −9.998273593045546208923007424049, −9.360905348949516554681540104257, −8.484729525105789790973146608444, −8.393143863540765287801842308182, −7.07681933404116959781280320400, −6.49286105348466055218357918045, −5.55232171485398205714244690090, −4.83572401632237409642686235671, −3.78385834662049451754732274084, −2.41374763992310282691552007523,
2.41374763992310282691552007523, 3.78385834662049451754732274084, 4.83572401632237409642686235671, 5.55232171485398205714244690090, 6.49286105348466055218357918045, 7.07681933404116959781280320400, 8.393143863540765287801842308182, 8.484729525105789790973146608444, 9.360905348949516554681540104257, 9.998273593045546208923007424049, 10.56134276430848544317017291752, 11.43511929194894990206894488011, 11.75826116113281771370809993152, 12.63529765841231317434736409625, 13.26957566008800311044646337672
