| L(s) = 1 | + 4-s + 9-s + 3·11-s + 16-s + 5·23-s − 14·31-s + 36-s − 5·37-s + 3·44-s − 5·47-s − 10·49-s + 5·53-s + 10·59-s + 64-s − 15·67-s − 11·71-s + 81-s + 15·89-s + 5·92-s + 5·97-s + 3·99-s − 30·103-s + 5·113-s − 2·121-s − 14·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 0.904·11-s + 1/4·16-s + 1.04·23-s − 2.51·31-s + 1/6·36-s − 0.821·37-s + 0.452·44-s − 0.729·47-s − 1.42·49-s + 0.686·53-s + 1.30·59-s + 1/8·64-s − 1.83·67-s − 1.30·71-s + 1/9·81-s + 1.58·89-s + 0.521·92-s + 0.507·97-s + 0.301·99-s − 2.95·103-s + 0.470·113-s − 0.181·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.35601624979104103183153876571, −7.04702436358865504789701678537, −6.59322271682898313357649648236, −6.23682392120206099446770046037, −5.73560264031448529475434020824, −5.23563412752963339046330701637, −4.94596375633482630110204765401, −4.29519404213312888866371685744, −3.79492138820006474858541834869, −3.43915872562403320206130548295, −2.94165555925975265899828448519, −2.21803249233456373097988150331, −1.63064248864045530329807281764, −1.22153238652211751193193043978, 0,
1.22153238652211751193193043978, 1.63064248864045530329807281764, 2.21803249233456373097988150331, 2.94165555925975265899828448519, 3.43915872562403320206130548295, 3.79492138820006474858541834869, 4.29519404213312888866371685744, 4.94596375633482630110204765401, 5.23563412752963339046330701637, 5.73560264031448529475434020824, 6.23682392120206099446770046037, 6.59322271682898313357649648236, 7.04702436358865504789701678537, 7.35601624979104103183153876571
