| L(s) = 1 | − 2·4-s − 2·5-s − 5·9-s − 2·13-s + 4·16-s − 8·17-s + 4·20-s − 7·25-s − 4·29-s + 10·36-s − 22·37-s + 20·41-s + 10·45-s − 10·49-s + 4·52-s + 4·53-s − 4·61-s − 8·64-s + 4·65-s + 16·68-s − 32·73-s − 8·80-s + 16·81-s + 16·85-s − 14·89-s − 26·97-s + 14·100-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 5/3·9-s − 0.554·13-s + 16-s − 1.94·17-s + 0.894·20-s − 7/5·25-s − 0.742·29-s + 5/3·36-s − 3.61·37-s + 3.12·41-s + 1.49·45-s − 1.42·49-s + 0.554·52-s + 0.549·53-s − 0.512·61-s − 64-s + 0.496·65-s + 1.94·68-s − 3.74·73-s − 0.894·80-s + 16/9·81-s + 1.73·85-s − 1.48·89-s − 2.63·97-s + 7/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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
−8.569451247588636860049749617820, −7.970223403211217946437728928671, −7.31285386083445627995852926185, −7.21661224338768342466420756206, −6.25972105104659001379078642019, −5.82135849339387357739621546932, −5.48444750282122420550662939059, −4.75768041533452306802121891710, −4.37350493013620998114647253846, −3.82887621952697788089432897883, −3.29304594675951575785139026233, −2.60654681549459182474671243206, −1.81431523402274649168122929883, 0, 0,
1.81431523402274649168122929883, 2.60654681549459182474671243206, 3.29304594675951575785139026233, 3.82887621952697788089432897883, 4.37350493013620998114647253846, 4.75768041533452306802121891710, 5.48444750282122420550662939059, 5.82135849339387357739621546932, 6.25972105104659001379078642019, 7.21661224338768342466420756206, 7.31285386083445627995852926185, 7.970223403211217946437728928671, 8.569451247588636860049749617820
