| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 4·9-s − 2·10-s + 16-s + 17-s − 4·18-s − 2·20-s − 7·25-s + 11·29-s + 32-s + 34-s − 4·36-s − 8·37-s − 2·40-s + 7·41-s + 8·45-s + 8·49-s − 7·50-s + 3·53-s + 11·58-s + 5·61-s + 64-s + 68-s − 4·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 4/3·9-s − 0.632·10-s + 1/4·16-s + 0.242·17-s − 0.942·18-s − 0.447·20-s − 7/5·25-s + 2.04·29-s + 0.176·32-s + 0.171·34-s − 2/3·36-s − 1.31·37-s − 0.316·40-s + 1.09·41-s + 1.19·45-s + 8/7·49-s − 0.989·50-s + 0.412·53-s + 1.44·58-s + 0.640·61-s + 1/8·64-s + 0.121·68-s − 0.471·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9020190524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9020190524\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 14 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
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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
−12.59553291146915511080661092877, −12.14914357965763811402258945951, −11.67666452745737778737148941465, −11.28615733627910955502391790681, −10.51284626564333131231254367811, −9.918635681762329691135585363506, −8.864568291064431286214969772691, −8.368363599824135263311855725638, −7.72105610635250465375639522353, −6.98295168141343444896521569686, −6.05386674990596023650926542265, −5.48493353587528670096291773113, −4.46988023346650659116902833531, −3.62568588139740439126212134267, −2.64765870112626908625429132935,
2.64765870112626908625429132935, 3.62568588139740439126212134267, 4.46988023346650659116902833531, 5.48493353587528670096291773113, 6.05386674990596023650926542265, 6.98295168141343444896521569686, 7.72105610635250465375639522353, 8.368363599824135263311855725638, 8.864568291064431286214969772691, 9.918635681762329691135585363506, 10.51284626564333131231254367811, 11.28615733627910955502391790681, 11.67666452745737778737148941465, 12.14914357965763811402258945951, 12.59553291146915511080661092877
