| L(s) = 1 | − 280·17-s − 140·25-s + 728·41-s − 348·49-s + 3.64e3·73-s + 2.18e3·89-s − 1.96e3·97-s + 3.64e3·113-s + 1.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.99·17-s − 1.11·25-s + 2.77·41-s − 1.01·49-s + 5.83·73-s + 2.60·89-s − 2.05·97-s + 3.03·113-s + 1.05·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.992014478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.992014478\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 14 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 174 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 58 T + p^{3} T^{2} )^{2}( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4074 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6958 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 754 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 33098 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 39814 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 141374 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 107294 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 282074 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 403998 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 399882 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 552526 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 703022 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 910 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 525278 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 632814 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 546 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.62459356827557796087651055847, −6.50934994326694564621360151637, −6.29787923646262909971415348053, −6.05830774221144873085812928053, −5.71921466156151293856638030712, −5.43000064196202396090976303485, −5.31008946437703544524167275836, −4.96347475837193427802990757268, −4.72019481021752679705175680594, −4.51196921198540720194854863944, −4.40899517226769976302220527345, −4.00552936213158347636781608641, −3.83455927725786060119868363253, −3.78728646948034556270840916867, −3.22418600486866050122847410967, −3.10430814164095505884132287060, −2.42838453537155661987102857042, −2.40982915273569669820312180834, −2.35189158228947013752104931061, −1.88337866719055649217695386256, −1.81569330885975492429446226552, −1.25198477479259634076190784347, −0.70515643916818248835163208312, −0.47231757526067068572610863332, −0.35945147363721191724828871066,
0.35945147363721191724828871066, 0.47231757526067068572610863332, 0.70515643916818248835163208312, 1.25198477479259634076190784347, 1.81569330885975492429446226552, 1.88337866719055649217695386256, 2.35189158228947013752104931061, 2.40982915273569669820312180834, 2.42838453537155661987102857042, 3.10430814164095505884132287060, 3.22418600486866050122847410967, 3.78728646948034556270840916867, 3.83455927725786060119868363253, 4.00552936213158347636781608641, 4.40899517226769976302220527345, 4.51196921198540720194854863944, 4.72019481021752679705175680594, 4.96347475837193427802990757268, 5.31008946437703544524167275836, 5.43000064196202396090976303485, 5.71921466156151293856638030712, 6.05830774221144873085812928053, 6.29787923646262909971415348053, 6.50934994326694564621360151637, 6.62459356827557796087651055847
