| L(s) = 1 | − 5-s + 4·7-s − 4·11-s + 2·13-s + 4·17-s + 8·19-s − 4·23-s + 2·29-s + 8·31-s − 4·35-s + 12·37-s + 6·41-s + 8·43-s − 4·47-s + 7·49-s + 12·53-s + 4·55-s + 4·59-s + 2·61-s − 2·65-s − 8·67-s − 12·73-s − 16·77-s + 16·83-s − 4·85-s − 12·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.97·37-s + 0.937·41-s + 1.21·43-s − 0.583·47-s + 49-s + 1.64·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.40·73-s − 1.82·77-s + 1.75·83-s − 0.433·85-s − 1.27·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.917298473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.917298473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.607775709437041602413722924695, −8.417825159260472172621727280129, −7.963075513021214427898954049499, −7.69466489956796322098432985878, −7.55791260785049701360181746942, −7.33388862660499371511840652318, −6.43646634818537288565510981220, −6.24450688203629144218535554065, −5.65567464688184135549705525501, −5.33975269808058419047303791955, −5.18120220181573496801746430151, −4.53974100949565747037044229012, −4.17352343707517017505178362686, −3.96394779183587688187157266762, −3.00087529291521224842535752278, −2.97768098757120242443489721192, −2.36936645781357900319827397737, −1.73186032262769934670712188038, −0.971950573948978149389944219909, −0.799323112237009899087073249450,
0.799323112237009899087073249450, 0.971950573948978149389944219909, 1.73186032262769934670712188038, 2.36936645781357900319827397737, 2.97768098757120242443489721192, 3.00087529291521224842535752278, 3.96394779183587688187157266762, 4.17352343707517017505178362686, 4.53974100949565747037044229012, 5.18120220181573496801746430151, 5.33975269808058419047303791955, 5.65567464688184135549705525501, 6.24450688203629144218535554065, 6.43646634818537288565510981220, 7.33388862660499371511840652318, 7.55791260785049701360181746942, 7.69466489956796322098432985878, 7.963075513021214427898954049499, 8.417825159260472172621727280129, 8.607775709437041602413722924695
