| L(s) = 1 | + 4-s + 4·5-s + 7-s − 2·9-s − 5·13-s + 16-s + 4·20-s − 6·23-s + 8·25-s + 28-s − 3·29-s + 4·35-s − 2·36-s − 8·45-s − 11·49-s − 5·52-s − 6·59-s − 2·63-s + 64-s − 20·65-s + 13·67-s + 6·71-s + 4·80-s − 5·81-s + 9·83-s − 5·91-s − 6·92-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s + 0.377·7-s − 2/3·9-s − 1.38·13-s + 1/4·16-s + 0.894·20-s − 1.25·23-s + 8/5·25-s + 0.188·28-s − 0.557·29-s + 0.676·35-s − 1/3·36-s − 1.19·45-s − 1.57·49-s − 0.693·52-s − 0.781·59-s − 0.251·63-s + 1/8·64-s − 2.48·65-s + 1.58·67-s + 0.712·71-s + 0.447·80-s − 5/9·81-s + 0.987·83-s − 0.524·91-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16820 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507833519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507833519\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
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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
−11.08661603835906627439652366747, −10.23858562194629731076232750078, −10.03971941445509995315194425412, −9.507275604298581857199210722511, −9.018079617612541473926938785192, −8.216386704595267551433915339737, −7.70696143831198272236901713458, −6.98461397487466204179882331116, −6.27991464987397779791610023531, −5.87788579500999534452013345660, −5.24096448944400160426705236449, −4.66638385389512071647400360849, −3.37106367684779932314518719379, −2.35746149861219306678210936929, −1.90854538214432964525538723189,
1.90854538214432964525538723189, 2.35746149861219306678210936929, 3.37106367684779932314518719379, 4.66638385389512071647400360849, 5.24096448944400160426705236449, 5.87788579500999534452013345660, 6.27991464987397779791610023531, 6.98461397487466204179882331116, 7.70696143831198272236901713458, 8.216386704595267551433915339737, 9.018079617612541473926938785192, 9.507275604298581857199210722511, 10.03971941445509995315194425412, 10.23858562194629731076232750078, 11.08661603835906627439652366747
