| L(s) = 1 | + 4·2-s + 33·5-s − 59·7-s − 64·8-s + 132·10-s + 147·11-s − 836·13-s − 236·14-s − 256·16-s + 2.16e3·17-s + 5.76e3·19-s + 588·22-s − 4.38e3·23-s + 3.12e3·25-s − 3.34e3·26-s + 1.86e3·29-s + 3.29e3·31-s + 8.64e3·34-s − 1.94e3·35-s − 7.91e3·37-s + 2.30e4·38-s − 2.11e3·40-s − 2.05e4·41-s + 8.77e3·43-s − 1.75e4·46-s + 1.26e4·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.590·5-s − 0.455·7-s − 0.353·8-s + 0.417·10-s + 0.366·11-s − 1.37·13-s − 0.321·14-s − 1/4·16-s + 1.81·17-s + 3.66·19-s + 0.259·22-s − 1.72·23-s + 25-s − 0.970·26-s + 0.412·29-s + 0.615·31-s + 1.28·34-s − 0.268·35-s − 0.950·37-s + 2.59·38-s − 0.208·40-s − 1.91·41-s + 0.723·43-s − 1.22·46-s + 0.836·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.403545552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.403545552\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 33 T - 2036 T^{2} - 33 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 59 T - 13326 T^{2} + 59 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 147 T - 139442 T^{2} - 147 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 836 T + 327603 T^{2} + 836 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 1080 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2882 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4386 T + 12800653 T^{2} + 4386 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1866 T - 17029193 T^{2} - 1866 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3295 T - 17772126 T^{2} - 3295 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3958 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 20586 T + 307927195 T^{2} + 20586 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 8770 T - 70095543 T^{2} - 8770 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12666 T - 68917451 T^{2} - 12666 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9621 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 21468 T - 254049275 T^{2} + 21468 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 36248 T + 469321203 T^{2} + 36248 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5174 T - 1323354831 T^{2} + 5174 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 63720 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 57953 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16448 T - 2806519695 T^{2} + 16448 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 69267 T + 858876646 T^{2} - 69267 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 54198 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 132961 T + 9091287264 T^{2} - 132961 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.24912232413008851916901901931, −12.01405424958203032566127084899, −11.59846318379651230988747959148, −10.45296910589291470827956769692, −10.08262418492647401242432798948, −9.805239162981618863750952203779, −9.346053657595192529851520448142, −8.755837004940307808988602449673, −7.81271173011976275383029850455, −7.44386872402677873226943646311, −7.02867873557787769238081929855, −6.04457582433655226190137373592, −5.69313326043327577605472983997, −5.12241672763431670360126467440, −4.70171803106711260878615624680, −3.41700858291232489875758478316, −3.36457622255857929345258191359, −2.48422259338145875140620687538, −1.38232643639721212474567771679, −0.66299855810703842519369931870,
0.66299855810703842519369931870, 1.38232643639721212474567771679, 2.48422259338145875140620687538, 3.36457622255857929345258191359, 3.41700858291232489875758478316, 4.70171803106711260878615624680, 5.12241672763431670360126467440, 5.69313326043327577605472983997, 6.04457582433655226190137373592, 7.02867873557787769238081929855, 7.44386872402677873226943646311, 7.81271173011976275383029850455, 8.755837004940307808988602449673, 9.346053657595192529851520448142, 9.805239162981618863750952203779, 10.08262418492647401242432798948, 10.45296910589291470827956769692, 11.59846318379651230988747959148, 12.01405424958203032566127084899, 12.24912232413008851916901901931
