| L(s) = 1 | − 3-s + 5-s + 4·7-s − 2·9-s − 11-s − 8·13-s − 15-s − 4·16-s − 4·21-s − 2·23-s − 4·25-s + 5·27-s + 14·31-s + 33-s + 4·35-s + 8·39-s + 16·41-s + 12·43-s − 2·45-s + 16·47-s + 4·48-s − 2·49-s − 12·53-s − 55-s − 8·63-s − 8·65-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s − 0.301·11-s − 2.21·13-s − 0.258·15-s − 16-s − 0.872·21-s − 0.417·23-s − 4/5·25-s + 0.962·27-s + 2.51·31-s + 0.174·33-s + 0.676·35-s + 1.28·39-s + 2.49·41-s + 1.82·43-s − 0.298·45-s + 2.33·47-s + 0.577·48-s − 2/7·49-s − 1.64·53-s − 0.134·55-s − 1.00·63-s − 0.992·65-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325407482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325407482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.845671750271850228494101894633, −8.276530745833509741483457469247, −7.78789607043645765017922373874, −7.56082953278459650643638402038, −7.06748750449497583777914546446, −6.23870064789214498217427813253, −5.99803243316135624496921094343, −5.41175071356617719099292784896, −4.90502322677283757606823458951, −4.44574490854809981150981932393, −4.29108769550776490491760393895, −2.69851752539611099704924785810, −2.63044898935838963010520851422, −1.91502190096627856445198269376, −0.68132836611621051670216891055,
0.68132836611621051670216891055, 1.91502190096627856445198269376, 2.63044898935838963010520851422, 2.69851752539611099704924785810, 4.29108769550776490491760393895, 4.44574490854809981150981932393, 4.90502322677283757606823458951, 5.41175071356617719099292784896, 5.99803243316135624496921094343, 6.23870064789214498217427813253, 7.06748750449497583777914546446, 7.56082953278459650643638402038, 7.78789607043645765017922373874, 8.276530745833509741483457469247, 8.845671750271850228494101894633
