| L(s) = 1 | − 2·5-s + 7-s + 3·9-s + 2·11-s + 7·13-s + 3·17-s + 6·19-s + 4·23-s − 7·25-s + 7·29-s + 8·31-s − 2·35-s − 9·37-s − 9·41-s − 10·43-s − 6·45-s − 4·47-s + 18·53-s − 4·55-s − 14·59-s + 5·61-s + 3·63-s − 14·65-s + 8·67-s − 10·71-s − 14·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 9-s + 0.603·11-s + 1.94·13-s + 0.727·17-s + 1.37·19-s + 0.834·23-s − 7/5·25-s + 1.29·29-s + 1.43·31-s − 0.338·35-s − 1.47·37-s − 1.40·41-s − 1.52·43-s − 0.894·45-s − 0.583·47-s + 2.47·53-s − 0.539·55-s − 1.82·59-s + 0.640·61-s + 0.377·63-s − 1.73·65-s + 0.977·67-s − 1.18·71-s − 1.63·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876086907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876086907\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.60559186698398431116290080569, −11.49862052475461955828631029153, −10.57187929245578525278766899395, −10.43117212952993506618170516742, −9.822632282135129075629780317203, −9.471408403704956456003547074650, −8.656021550420366890114093580093, −8.319069613780472477798336206496, −8.127301024352791402083729070614, −7.38177118936365565035624442487, −6.81533664913621079047290992508, −6.68977739888406436917178181352, −5.69796008771547316423098302330, −5.41825273629224614893122090096, −4.46066407150155076441769736227, −4.20912539731422956209975365107, −3.36039004425604906387190846365, −3.21778095408038641361974355011, −1.60441124135414608045690655263, −1.14656171956604921705932898900,
1.14656171956604921705932898900, 1.60441124135414608045690655263, 3.21778095408038641361974355011, 3.36039004425604906387190846365, 4.20912539731422956209975365107, 4.46066407150155076441769736227, 5.41825273629224614893122090096, 5.69796008771547316423098302330, 6.68977739888406436917178181352, 6.81533664913621079047290992508, 7.38177118936365565035624442487, 8.127301024352791402083729070614, 8.319069613780472477798336206496, 8.656021550420366890114093580093, 9.471408403704956456003547074650, 9.822632282135129075629780317203, 10.43117212952993506618170516742, 10.57187929245578525278766899395, 11.49862052475461955828631029153, 11.60559186698398431116290080569
