| L(s) = 1 | + 14·7-s − 20·23-s + 22·25-s + 147·49-s − 220·71-s − 260·79-s + 52·113-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 280·161-s + 163-s + 167-s + 310·169-s + 173-s + 308·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·7-s − 0.869·23-s + 0.879·25-s + 3·49-s − 3.09·71-s − 3.29·79-s + 0.460·113-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 1.73·161-s + 0.00613·163-s + 0.00598·167-s + 1.83·169-s + 0.00578·173-s + 1.75·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.704253636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.704253636\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 22 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 310 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 650 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7370 T^{2} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13130 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−8.905413955180413903731995308448, −8.679437747347805917702158442653, −8.417610953695573731439594829375, −8.142162648135797549346924437916, −7.41049543057399357203140370751, −7.39744139844505844982967995423, −7.05254454948978585113206874928, −6.31084852570963543737342631501, −5.86964000995592944723267348662, −5.60421667357358108436214468828, −5.12633117194320711491560355762, −4.61720476533318288255584822129, −4.30760384176978428993386361777, −4.10487603039135096539346078530, −3.13110392646624773242943929686, −2.91757824631925370332415185022, −2.05597076611335578342927298225, −1.76707012980805688642068785522, −1.22436656333937190535166331034, −0.50606179316495413297617823634,
0.50606179316495413297617823634, 1.22436656333937190535166331034, 1.76707012980805688642068785522, 2.05597076611335578342927298225, 2.91757824631925370332415185022, 3.13110392646624773242943929686, 4.10487603039135096539346078530, 4.30760384176978428993386361777, 4.61720476533318288255584822129, 5.12633117194320711491560355762, 5.60421667357358108436214468828, 5.86964000995592944723267348662, 6.31084852570963543737342631501, 7.05254454948978585113206874928, 7.39744139844505844982967995423, 7.41049543057399357203140370751, 8.142162648135797549346924437916, 8.417610953695573731439594829375, 8.679437747347805917702158442653, 8.905413955180413903731995308448
