| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 3·9-s − 10·11-s + 13-s + 2·15-s + 2·17-s − 4·19-s + 4·21-s − 2·23-s − 2·25-s + 4·27-s − 10·29-s − 4·31-s − 20·33-s + 2·35-s + 2·39-s − 2·41-s − 5·43-s + 3·45-s − 10·47-s + 3·49-s + 4·51-s − 5·53-s − 10·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 9-s − 3.01·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.417·23-s − 2/5·25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 3.48·33-s + 0.338·35-s + 0.320·39-s − 0.312·41-s − 0.762·43-s + 0.447·45-s − 1.45·47-s + 3/7·49-s + 0.560·51-s − 0.686·53-s − 1.34·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 85 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 127 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 107 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 127 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.77101008325541605634216473251, −7.48988892228199219276017745353, −7.08573969509191910202644796491, −6.95917894497327301570283606295, −6.11557847814457826750481680418, −5.94601327157683506046132700998, −5.47664138694075781671457427360, −5.29537235119669112638215405830, −4.82928599466992678616499789068, −4.65614016252587385015870264288, −3.90937607811772086584339063382, −3.76431188425055562228939000717, −3.23039397008500129541672834323, −2.83044343101377629766332249768, −2.36118927780005340615701164414, −2.17530800916773690356540313391, −1.71649162824622335529075221087, −1.28485741823958006760553428683, 0, 0,
1.28485741823958006760553428683, 1.71649162824622335529075221087, 2.17530800916773690356540313391, 2.36118927780005340615701164414, 2.83044343101377629766332249768, 3.23039397008500129541672834323, 3.76431188425055562228939000717, 3.90937607811772086584339063382, 4.65614016252587385015870264288, 4.82928599466992678616499789068, 5.29537235119669112638215405830, 5.47664138694075781671457427360, 5.94601327157683506046132700998, 6.11557847814457826750481680418, 6.95917894497327301570283606295, 7.08573969509191910202644796491, 7.48988892228199219276017745353, 7.77101008325541605634216473251
