| L(s) = 1 | + 4·5-s + 8·11-s + 8·19-s + 11·25-s + 8·29-s − 16·41-s − 2·49-s + 32·55-s + 24·59-s − 4·61-s + 16·71-s − 16·79-s + 32·95-s − 24·101-s + 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.41·11-s + 1.83·19-s + 11/5·25-s + 1.48·29-s − 2.49·41-s − 2/7·49-s + 4.31·55-s + 3.12·59-s − 0.512·61-s + 1.89·71-s − 1.80·79-s + 3.28·95-s − 2.38·101-s + 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.209192634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.209192634\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.829575609809099927549616554996, −8.703181825472352089130206028342, −8.490297534402433276537211892736, −7.899169213253494066929798070210, −7.07951219244992149513245311167, −7.06789268693692973831890982023, −6.53519549069998585644786240805, −6.53410898244224909583266021305, −5.75146765619206701106438276300, −5.71528983848918644149208669405, −5.07354328575928649516733747845, −4.85726124351368323471911354036, −4.26810741495646295736502979548, −3.69898209957509240611984397351, −3.32604611750637173917580441461, −2.90955782878987105188415109424, −2.19328313748489042836202646523, −1.78332920920835719852468907493, −1.15237084760825592137910308431, −0.972136604465642133399659551095,
0.972136604465642133399659551095, 1.15237084760825592137910308431, 1.78332920920835719852468907493, 2.19328313748489042836202646523, 2.90955782878987105188415109424, 3.32604611750637173917580441461, 3.69898209957509240611984397351, 4.26810741495646295736502979548, 4.85726124351368323471911354036, 5.07354328575928649516733747845, 5.71528983848918644149208669405, 5.75146765619206701106438276300, 6.53410898244224909583266021305, 6.53519549069998585644786240805, 7.06789268693692973831890982023, 7.07951219244992149513245311167, 7.899169213253494066929798070210, 8.490297534402433276537211892736, 8.703181825472352089130206028342, 8.829575609809099927549616554996
