| L(s) = 1 | + 3·2-s + 3·4-s − 7-s − 3·8-s + 6·9-s − 2·11-s − 3·14-s − 13·16-s + 18·18-s − 6·22-s + 3·23-s + 3·25-s − 3·28-s + 9·29-s − 15·32-s + 18·36-s + 3·37-s − 7·43-s − 6·44-s + 9·46-s − 6·49-s + 9·50-s + 53-s + 3·56-s + 27·58-s − 6·63-s + 3·64-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 0.377·7-s − 1.06·8-s + 2·9-s − 0.603·11-s − 0.801·14-s − 3.25·16-s + 4.24·18-s − 1.27·22-s + 0.625·23-s + 3/5·25-s − 0.566·28-s + 1.67·29-s − 2.65·32-s + 3·36-s + 0.493·37-s − 1.06·43-s − 0.904·44-s + 1.32·46-s − 6/7·49-s + 1.27·50-s + 0.137·53-s + 0.400·56-s + 3.54·58-s − 0.755·63-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357259 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357259 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.717854596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.717854596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 317 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−8.735106820132059306990210400063, −8.151098661206566778989506991594, −7.71615185410232343819509390669, −6.89009883742859003900104306790, −6.69595829291047286311971650786, −6.38889353700107083054347936974, −5.64337094440734635031855053349, −5.07316722243699056969792369726, −4.79135874265832392188210652530, −4.48947946319284949572076974052, −3.82651701907696595154129210041, −3.44842937231588178921238287278, −2.85592362480477021827284561635, −2.14221416372440871582045593033, −0.921700946204613052563643481869,
0.921700946204613052563643481869, 2.14221416372440871582045593033, 2.85592362480477021827284561635, 3.44842937231588178921238287278, 3.82651701907696595154129210041, 4.48947946319284949572076974052, 4.79135874265832392188210652530, 5.07316722243699056969792369726, 5.64337094440734635031855053349, 6.38889353700107083054347936974, 6.69595829291047286311971650786, 6.89009883742859003900104306790, 7.71615185410232343819509390669, 8.151098661206566778989506991594, 8.735106820132059306990210400063
