| L(s) = 1 | + 6·11-s − 8·13-s − 2·25-s − 4·37-s − 24·47-s + 2·49-s + 30·59-s − 16·61-s + 12·71-s − 22·73-s − 24·83-s + 26·97-s + 6·107-s − 8·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.80·11-s − 2.21·13-s − 2/5·25-s − 0.657·37-s − 3.50·47-s + 2/7·49-s + 3.90·59-s − 2.04·61-s + 1.42·71-s − 2.57·73-s − 2.63·83-s + 2.63·97-s + 0.580·107-s − 0.766·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099725332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099725332\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.544146853427248120847010298072, −7.909104865861819619266446913079, −7.67585769261164859954415447825, −7.25341786170802775526048816972, −6.89657598497463148020203748383, −6.77458589464817829041759209172, −6.13131401129052213773272265564, −6.12117267776560531834819161570, −5.31104750084759759343467302559, −5.04639152747543618677694527593, −4.88696217949445653420382019763, −4.12315867435088967987816226232, −4.10920124358460771793855052000, −3.54946211292844372993121295468, −2.98857828378875688367914182606, −2.71389588890699530735884162776, −2.00511844363563346756604057847, −1.72530005874072464053233690917, −1.15861876444581768248821836592, −0.28535962022084670434016196853,
0.28535962022084670434016196853, 1.15861876444581768248821836592, 1.72530005874072464053233690917, 2.00511844363563346756604057847, 2.71389588890699530735884162776, 2.98857828378875688367914182606, 3.54946211292844372993121295468, 4.10920124358460771793855052000, 4.12315867435088967987816226232, 4.88696217949445653420382019763, 5.04639152747543618677694527593, 5.31104750084759759343467302559, 6.12117267776560531834819161570, 6.13131401129052213773272265564, 6.77458589464817829041759209172, 6.89657598497463148020203748383, 7.25341786170802775526048816972, 7.67585769261164859954415447825, 7.909104865861819619266446913079, 8.544146853427248120847010298072
