| L(s) = 1 | + 8·23-s − 8·25-s + 12·29-s − 8·37-s + 8·43-s + 8·53-s + 8·67-s + 24·71-s + 16·79-s + 32·107-s − 8·109-s + 32·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.66·23-s − 8/5·25-s + 2.22·29-s − 1.31·37-s + 1.21·43-s + 1.09·53-s + 0.977·67-s + 2.84·71-s + 1.80·79-s + 3.09·107-s − 0.766·109-s + 3.01·113-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 + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.211796970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.211796970\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−8.657284465829937159154535871358, −8.479444863410698432098661650243, −7.898343611211250987278091707209, −7.82188852888362006324535289154, −7.17495818684444567627314517359, −6.95208894495956377479898243742, −6.43267427097012000844105657215, −6.37388625516611858227023080884, −5.58192272729343979893803934832, −5.46697428890159710031616912384, −4.82833326096884863195989945874, −4.76441682845476448278270134228, −3.92842572480601102662905257371, −3.87140700231365201649265683325, −3.11382086277298159982717652546, −2.92312634098921623400413298936, −2.04511721880421838419872096940, −2.04358096292737612482560710779, −0.898270004494159331583658136594, −0.70492836926337751688275658047,
0.70492836926337751688275658047, 0.898270004494159331583658136594, 2.04358096292737612482560710779, 2.04511721880421838419872096940, 2.92312634098921623400413298936, 3.11382086277298159982717652546, 3.87140700231365201649265683325, 3.92842572480601102662905257371, 4.76441682845476448278270134228, 4.82833326096884863195989945874, 5.46697428890159710031616912384, 5.58192272729343979893803934832, 6.37388625516611858227023080884, 6.43267427097012000844105657215, 6.95208894495956377479898243742, 7.17495818684444567627314517359, 7.82188852888362006324535289154, 7.898343611211250987278091707209, 8.479444863410698432098661650243, 8.657284465829937159154535871358
