| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 2·9-s − 4·14-s + 16-s + 2·17-s − 2·18-s + 9·23-s − 2·25-s + 4·28-s + 8·31-s − 32-s − 2·34-s + 2·36-s − 4·41-s − 9·46-s + 47-s − 2·49-s + 2·50-s − 4·56-s − 8·62-s + 8·63-s + 64-s + 2·68-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 2/3·9-s − 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.87·23-s − 2/5·25-s + 0.755·28-s + 1.43·31-s − 0.176·32-s − 0.342·34-s + 1/3·36-s − 0.624·41-s − 1.32·46-s + 0.145·47-s − 2/7·49-s + 0.282·50-s − 0.534·56-s − 1.01·62-s + 1.00·63-s + 1/8·64-s + 0.242·68-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584195372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584195372\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.423429927226921162732899514419, −8.722613975537749195839023650009, −8.243781316607903061482910044854, −8.133697454065306009861698322898, −7.43999882001361156014161374978, −6.96076004104489026118803709960, −6.65237566954454125795067283023, −5.75728559315769138278852517682, −5.28098919401238598744212693744, −4.68943648242487004282703228915, −4.27894624506524931748538513911, −3.31833266716801866946896291648, −2.64131017557819879765966837572, −1.67168184853292857301205758313, −1.11479969868756858575990569945,
1.11479969868756858575990569945, 1.67168184853292857301205758313, 2.64131017557819879765966837572, 3.31833266716801866946896291648, 4.27894624506524931748538513911, 4.68943648242487004282703228915, 5.28098919401238598744212693744, 5.75728559315769138278852517682, 6.65237566954454125795067283023, 6.96076004104489026118803709960, 7.43999882001361156014161374978, 8.133697454065306009861698322898, 8.243781316607903061482910044854, 8.722613975537749195839023650009, 9.423429927226921162732899514419
