| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 5·9-s − 14-s + 16-s − 9·17-s + 5·18-s − 3·23-s + 2·25-s + 28-s + 4·31-s − 32-s + 9·34-s − 5·36-s − 9·41-s + 3·46-s − 9·47-s + 49-s − 2·50-s − 56-s − 4·62-s − 5·63-s + 64-s − 9·68-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 5/3·9-s − 0.267·14-s + 1/4·16-s − 2.18·17-s + 1.17·18-s − 0.625·23-s + 2/5·25-s + 0.188·28-s + 0.718·31-s − 0.176·32-s + 1.54·34-s − 5/6·36-s − 1.40·41-s + 0.442·46-s − 1.31·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.508·62-s − 0.629·63-s + 1/8·64-s − 1.09·68-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−10.16503550298585820097002192537, −9.183105464166945810672815012539, −8.803492526446667316711688403013, −8.565983171959853283933569857530, −8.063621987925375588879513174290, −7.39421188716369306557033815179, −6.70885443291402377233247170593, −6.26828606774537762458311499295, −5.73771297527460382349868991289, −4.92023477868249713444324566679, −4.39611675489762487688262311461, −3.31155823054676521760797309766, −2.63959424907761678348027679278, −1.83998833486204604502597426161, 0,
1.83998833486204604502597426161, 2.63959424907761678348027679278, 3.31155823054676521760797309766, 4.39611675489762487688262311461, 4.92023477868249713444324566679, 5.73771297527460382349868991289, 6.26828606774537762458311499295, 6.70885443291402377233247170593, 7.39421188716369306557033815179, 8.063621987925375588879513174290, 8.565983171959853283933569857530, 8.803492526446667316711688403013, 9.183105464166945810672815012539, 10.16503550298585820097002192537
