| L(s) = 1 | − 2·4-s − 4·5-s + 4·16-s + 8·20-s − 40·23-s − 38·25-s + 24·31-s + 60·37-s − 88·47-s + 98·49-s + 112·53-s + 160·59-s − 8·64-s + 160·67-s − 120·71-s − 16·80-s − 160·89-s + 80·92-s + 76·100-s + 280·103-s − 420·113-s + 160·115-s − 48·124-s + 268·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 4/5·5-s + 1/4·16-s + 2/5·20-s − 1.73·23-s − 1.51·25-s + 0.774·31-s + 1.62·37-s − 1.87·47-s + 2·49-s + 2.11·53-s + 2.71·59-s − 1/8·64-s + 2.38·67-s − 1.69·71-s − 1/5·80-s − 1.79·89-s + 0.869·92-s + 0.759·100-s + 2.71·103-s − 3.71·113-s + 1.39·115-s − 0.387·124-s + 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.371493232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.371493232\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 288 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 690 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1632 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2912 T^{2} + p^{4} T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6560 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4608 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8610 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12210 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.969818554140440162278916876435, −8.625593433711442421106288728213, −8.158786690531956967606404472334, −8.066320907792031555105893492492, −7.67193078996614117000153011086, −7.16874005911068688823828189274, −6.85515526324869800403421900367, −6.21971252164068912269715588324, −5.98822931140637687662345616491, −5.36661107258802145237724757859, −5.29846107645755582425162532913, −4.32972824216435895872389043960, −4.25606322571319941957927212094, −3.82126758632687554473736553103, −3.57140300776507165156957576388, −2.62241548278392226037990282782, −2.39444376980377885464837010925, −1.71002423345057271733279239564, −0.863512086040662833125563929160, −0.37707185539982901487170741712,
0.37707185539982901487170741712, 0.863512086040662833125563929160, 1.71002423345057271733279239564, 2.39444376980377885464837010925, 2.62241548278392226037990282782, 3.57140300776507165156957576388, 3.82126758632687554473736553103, 4.25606322571319941957927212094, 4.32972824216435895872389043960, 5.29846107645755582425162532913, 5.36661107258802145237724757859, 5.98822931140637687662345616491, 6.21971252164068912269715588324, 6.85515526324869800403421900367, 7.16874005911068688823828189274, 7.67193078996614117000153011086, 8.066320907792031555105893492492, 8.158786690531956967606404472334, 8.625593433711442421106288728213, 8.969818554140440162278916876435
