| L(s) = 1 | − 8·11-s − 12·13-s + 12·23-s + 10·25-s − 12·37-s − 12·47-s − 4·49-s + 8·59-s + 12·61-s − 12·71-s + 12·73-s − 32·83-s − 24·97-s − 32·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 96·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 3.32·13-s + 2.50·23-s + 2·25-s − 1.97·37-s − 1.75·47-s − 4/7·49-s + 1.04·59-s + 1.53·61-s − 1.42·71-s + 1.40·73-s − 3.51·83-s − 2.43·97-s − 3.09·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.02·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.184119526784080096930054583337, −7.76714131083724239901865740792, −7.28663558619794145335692789488, −7.02491795510718615019431936981, −6.83405135447474564037169283022, −6.72830861693029482285171102857, −5.59738193717085894569396855219, −5.30809244850357323314924184756, −5.21799090855866861698966030986, −4.96981545298691664253380339263, −4.61752563126245202166051772557, −4.16502054511091565788199569360, −3.11753878681541518270339046385, −3.08596394900680291147734248050, −2.68276431274612271923823533354, −2.43384670690819110329111151351, −1.78827932214867660260249596332, −1.01465676791758919550831411091, 0, 0,
1.01465676791758919550831411091, 1.78827932214867660260249596332, 2.43384670690819110329111151351, 2.68276431274612271923823533354, 3.08596394900680291147734248050, 3.11753878681541518270339046385, 4.16502054511091565788199569360, 4.61752563126245202166051772557, 4.96981545298691664253380339263, 5.21799090855866861698966030986, 5.30809244850357323314924184756, 5.59738193717085894569396855219, 6.72830861693029482285171102857, 6.83405135447474564037169283022, 7.02491795510718615019431936981, 7.28663558619794145335692789488, 7.76714131083724239901865740792, 8.184119526784080096930054583337
