| L(s) = 1 | + 2·3-s + 4-s − 4·5-s + 3·9-s − 4·11-s + 2·12-s − 8·15-s + 16-s − 4·20-s + 2·25-s + 4·27-s + 16·31-s − 8·33-s + 3·36-s − 4·37-s − 4·44-s − 12·45-s + 2·48-s − 14·49-s + 12·53-s + 16·55-s + 24·59-s − 8·60-s + 64-s − 24·67-s + 4·75-s − 4·80-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 1.78·5-s + 9-s − 1.20·11-s + 0.577·12-s − 2.06·15-s + 1/4·16-s − 0.894·20-s + 2/5·25-s + 0.769·27-s + 2.87·31-s − 1.39·33-s + 1/2·36-s − 0.657·37-s − 0.603·44-s − 1.78·45-s + 0.288·48-s − 2·49-s + 1.64·53-s + 2.15·55-s + 3.12·59-s − 1.03·60-s + 1/8·64-s − 2.93·67-s + 0.461·75-s − 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1258884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1258884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.958882066985845549410362974675, −7.36440720073199929079936076774, −7.17015458110489899878368281457, −6.72884801535058667170696059539, −6.11887476554628092067039386855, −5.51457882926074302977186886250, −4.98149691236367693223181594270, −4.33709791955016592854024404044, −4.13875090005906653574798019221, −3.57025483924234576580912047342, −2.92580973222471519775314024765, −2.76309813409348720224135008444, −2.04715104709696587907765661420, −1.10797907566740122108357243035, 0,
1.10797907566740122108357243035, 2.04715104709696587907765661420, 2.76309813409348720224135008444, 2.92580973222471519775314024765, 3.57025483924234576580912047342, 4.13875090005906653574798019221, 4.33709791955016592854024404044, 4.98149691236367693223181594270, 5.51457882926074302977186886250, 6.11887476554628092067039386855, 6.72884801535058667170696059539, 7.17015458110489899878368281457, 7.36440720073199929079936076774, 7.958882066985845549410362974675
