| L(s) = 1 | − 4·5-s − 10·13-s + 10·17-s + 11·25-s + 10·37-s − 16·41-s − 10·53-s − 24·61-s + 40·65-s + 10·73-s − 40·85-s + 10·97-s − 4·101-s − 30·113-s + 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2.77·13-s + 2.42·17-s + 11/5·25-s + 1.64·37-s − 2.49·41-s − 1.37·53-s − 3.07·61-s + 4.96·65-s + 1.17·73-s − 4.33·85-s + 1.01·97-s − 0.398·101-s − 2.82·113-s + 2·121-s − 2.14·125-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 + 3.84·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6853867155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6853867155\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.64065210830210825438694943800, −10.13501475744345699642418941512, −9.715288814539549451072271140271, −9.541155496830400545204402493216, −8.921850869445639689069105511047, −8.147513325142824767473202160957, −7.968249838097597296014444644572, −7.65505867975109223610531894634, −7.31861043615295832735158602451, −6.90350327093729244211903579518, −6.28591592061112047681363820206, −5.56676128846362329260603883788, −5.01419410160804988396575696349, −4.75168108136968396186086502134, −4.28295712761428366738200999132, −3.44409810353912618084281220687, −3.17301110087739539798135539875, −2.63212306855336792251199475316, −1.56151718046754628350246714915, −0.43842981696271729148507804072,
0.43842981696271729148507804072, 1.56151718046754628350246714915, 2.63212306855336792251199475316, 3.17301110087739539798135539875, 3.44409810353912618084281220687, 4.28295712761428366738200999132, 4.75168108136968396186086502134, 5.01419410160804988396575696349, 5.56676128846362329260603883788, 6.28591592061112047681363820206, 6.90350327093729244211903579518, 7.31861043615295832735158602451, 7.65505867975109223610531894634, 7.968249838097597296014444644572, 8.147513325142824767473202160957, 8.921850869445639689069105511047, 9.541155496830400545204402493216, 9.715288814539549451072271140271, 10.13501475744345699642418941512, 10.64065210830210825438694943800
