L(s) = 1 | − 4·11-s − 12·23-s − 2·25-s − 8·29-s − 4·37-s + 8·43-s + 24·53-s + 24·67-s + 12·71-s − 16·79-s − 36·107-s + 28·109-s − 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 2.50·23-s − 2/5·25-s − 1.48·29-s − 0.657·37-s + 1.21·43-s + 3.29·53-s + 2.93·67-s + 1.42·71-s − 1.80·79-s − 3.48·107-s + 2.68·109-s − 1.50·113-s − 0.909·121-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 + 6/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.419052970\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419052970\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−8.191245684407558222333873557556, −7.76176467193206468729065996394, −7.44542629858281685930772452608, −7.12536027614820631411796659282, −6.79378639546782859859020244724, −6.39114218894079728637947381454, −5.73614789538599436254478027254, −5.67286579255758491394791769863, −5.48821545586914128176805085498, −5.07414559061215153961500792587, −4.31304508683590084957154987813, −4.24343111771904064186306611382, −3.65653234180789986182151587459, −3.63527965874330099060081154048, −2.78451422221460853181820425039, −2.47589361776507879777210754850, −1.93617327452298501371290145868, −1.90251830199611206592029568265, −0.853915897345836822854217443224, −0.34662703793782673853781457279,
0.34662703793782673853781457279, 0.853915897345836822854217443224, 1.90251830199611206592029568265, 1.93617327452298501371290145868, 2.47589361776507879777210754850, 2.78451422221460853181820425039, 3.63527965874330099060081154048, 3.65653234180789986182151587459, 4.24343111771904064186306611382, 4.31304508683590084957154987813, 5.07414559061215153961500792587, 5.48821545586914128176805085498, 5.67286579255758491394791769863, 5.73614789538599436254478027254, 6.39114218894079728637947381454, 6.79378639546782859859020244724, 7.12536027614820631411796659282, 7.44542629858281685930772452608, 7.76176467193206468729065996394, 8.191245684407558222333873557556