| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 14-s + 2·15-s − 16-s + 17-s + 18-s − 21-s − 24-s + 25-s + 2·27-s + 2·30-s − 4·31-s + 34-s − 2·35-s − 37-s − 2·40-s − 42-s + 43-s + 2·45-s + 2·47-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 14-s + 2·15-s − 16-s + 17-s + 18-s − 21-s − 24-s + 25-s + 2·27-s + 2·30-s − 4·31-s + 34-s − 2·35-s − 37-s − 2·40-s − 42-s + 43-s + 2·45-s + 2·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.868757836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868757836\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.694424961088856943637123830944, −9.656361258447832361351970541212, −9.176236979548475948595703556973, −8.901489103286226464715055778822, −8.821906459645765264092299128194, −7.85641125598285750583854574681, −7.55898338060417101902618265648, −6.95082644074089100118944442157, −6.73673343551487987721213397314, −6.13245980298341720261497565501, −5.66789066664759067689681095834, −5.40223010768282246688390057095, −5.32785995138195198531269442945, −4.16073480727035636401139162302, −4.11916395240540342828665887712, −3.29754264747014878341025719717, −3.19189314768370573786091537147, −2.36510886821755244404225231322, −2.09984144407098855652649634303, −1.33863083999387774122270682158,
1.33863083999387774122270682158, 2.09984144407098855652649634303, 2.36510886821755244404225231322, 3.19189314768370573786091537147, 3.29754264747014878341025719717, 4.11916395240540342828665887712, 4.16073480727035636401139162302, 5.32785995138195198531269442945, 5.40223010768282246688390057095, 5.66789066664759067689681095834, 6.13245980298341720261497565501, 6.73673343551487987721213397314, 6.95082644074089100118944442157, 7.55898338060417101902618265648, 7.85641125598285750583854574681, 8.821906459645765264092299128194, 8.901489103286226464715055778822, 9.176236979548475948595703556973, 9.656361258447832361351970541212, 9.694424961088856943637123830944
