| L(s) = 1 | − 4-s − 4·5-s − 4·7-s + 16-s − 4·17-s + 4·19-s + 4·20-s + 11·25-s + 4·28-s − 14·29-s − 8·31-s + 16·35-s − 12·37-s − 4·47-s + 8·49-s − 2·53-s + 4·59-s + 2·61-s − 64-s + 4·68-s + 24·71-s − 10·73-s − 4·76-s + 24·79-s − 4·80-s − 9·81-s + 8·83-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 1.51·7-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.755·28-s − 2.59·29-s − 1.43·31-s + 2.70·35-s − 1.97·37-s − 0.583·47-s + 8/7·49-s − 0.274·53-s + 0.520·59-s + 0.256·61-s − 1/8·64-s + 0.485·68-s + 2.84·71-s − 1.17·73-s − 0.458·76-s + 2.70·79-s − 0.447·80-s − 81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2766054425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2766054425\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−11.52408095105280412824585195220, −11.05837547175172259104115780760, −11.01953573661147156632064910491, −10.25618407552202437619726152264, −9.623516144900915431462891346675, −9.258357554722968770592435204692, −9.050472398493849135696134044775, −8.226410853831396909943080683464, −8.049025178886690148935666509297, −7.22383955773552856424802437728, −6.99996654914562854651426053592, −6.68162990850165074635186615510, −5.52774474511993154997395398781, −5.48336745880939156782373721048, −4.52095980663305453796274492821, −3.94763464741997591106777175486, −3.38876272255592423234517035955, −3.31976316621932133557067001860, −1.99332978472701014131446014688, −0.34922906211715481459469904070,
0.34922906211715481459469904070, 1.99332978472701014131446014688, 3.31976316621932133557067001860, 3.38876272255592423234517035955, 3.94763464741997591106777175486, 4.52095980663305453796274492821, 5.48336745880939156782373721048, 5.52774474511993154997395398781, 6.68162990850165074635186615510, 6.99996654914562854651426053592, 7.22383955773552856424802437728, 8.049025178886690148935666509297, 8.226410853831396909943080683464, 9.050472398493849135696134044775, 9.258357554722968770592435204692, 9.623516144900915431462891346675, 10.25618407552202437619726152264, 11.01953573661147156632064910491, 11.05837547175172259104115780760, 11.52408095105280412824585195220
