| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s + 9-s + 2·14-s − 16-s + 10·17-s + 18-s − 6·23-s + 8·25-s − 2·28-s − 10·31-s + 5·32-s + 10·34-s − 36-s + 4·41-s − 6·46-s + 6·47-s − 3·49-s + 8·50-s − 6·56-s − 10·62-s + 2·63-s + 7·64-s − 10·68-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.534·14-s − 1/4·16-s + 2.42·17-s + 0.235·18-s − 1.25·23-s + 8/5·25-s − 0.377·28-s − 1.79·31-s + 0.883·32-s + 1.71·34-s − 1/6·36-s + 0.624·41-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + 1.13·50-s − 0.801·56-s − 1.27·62-s + 0.251·63-s + 7/8·64-s − 1.21·68-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.634377899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634377899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.63741207941122620867037340602, −9.957755573628363529546213010401, −9.617657053762168439220202869179, −8.973759465922080274841874472885, −8.427677052873413924134047785712, −7.80051888061486073751659237520, −7.48399895733211269670022567674, −6.61556205600024921018892553038, −5.85672787782641584026137936801, −5.35119619758397882671338909656, −4.97860560806729834090941136744, −4.04850914911787082610730307448, −3.62166430869030046103953554477, −2.71045259179536113161536201563, −1.32394996127883385330887628640,
1.32394996127883385330887628640, 2.71045259179536113161536201563, 3.62166430869030046103953554477, 4.04850914911787082610730307448, 4.97860560806729834090941136744, 5.35119619758397882671338909656, 5.85672787782641584026137936801, 6.61556205600024921018892553038, 7.48399895733211269670022567674, 7.80051888061486073751659237520, 8.427677052873413924134047785712, 8.973759465922080274841874472885, 9.617657053762168439220202869179, 9.957755573628363529546213010401, 10.63741207941122620867037340602
