| L(s) = 1 | + 6·7-s − 2·23-s + 12·29-s + 18·43-s − 14·47-s + 18·49-s + 6·67-s + 22·83-s − 12·89-s + 36·101-s + 18·103-s + 26·107-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 12·161-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.26·7-s − 0.417·23-s + 2.22·29-s + 2.74·43-s − 2.04·47-s + 18/7·49-s + 0.733·67-s + 2.41·83-s − 1.27·89-s + 3.58·101-s + 1.77·103-s + 2.51·107-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.945·161-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.759566818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.759566818\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.096559439968840208930334390025, −7.83635595286014022258095023805, −7.38196287531328426459859164677, −7.33821538064769685917881881518, −6.55304506873024720074428570896, −6.45660213987737243046529437775, −5.94103262162492931038179789181, −5.67834446320787553307869061323, −5.07503917300434549897758671992, −4.87903609808944360686114794223, −4.58767631266704864393775510878, −4.40062423072192247189511174106, −3.76033628270095452825522072513, −3.44818114533106543883266761275, −2.79107918921604666135985678376, −2.46129744287166072400837008542, −1.84434307512047331684423106027, −1.75800995098082718900107585275, −0.869168304842140774857568928108, −0.75755906389362835296778853913,
0.75755906389362835296778853913, 0.869168304842140774857568928108, 1.75800995098082718900107585275, 1.84434307512047331684423106027, 2.46129744287166072400837008542, 2.79107918921604666135985678376, 3.44818114533106543883266761275, 3.76033628270095452825522072513, 4.40062423072192247189511174106, 4.58767631266704864393775510878, 4.87903609808944360686114794223, 5.07503917300434549897758671992, 5.67834446320787553307869061323, 5.94103262162492931038179789181, 6.45660213987737243046529437775, 6.55304506873024720074428570896, 7.33821538064769685917881881518, 7.38196287531328426459859164677, 7.83635595286014022258095023805, 8.096559439968840208930334390025
