| L(s) = 1 | + 617·3-s + 8.19e3·4-s + 2.31e5·7-s + 5.31e5·9-s + 3.19e6·11-s + 5.05e6·12-s + 5.03e7·16-s + 3.06e7·17-s + 1.42e8·21-s − 4.62e8·23-s + 4.88e8·25-s + 7.48e8·27-s + 1.89e9·28-s − 9.91e8·29-s + 1.01e8·31-s + 1.97e9·33-s + 4.35e9·36-s + 5.01e9·37-s − 1.89e10·41-s + 2.62e10·44-s + 3.10e10·48-s + 1.38e10·49-s + 1.88e10·51-s + 4.86e10·59-s − 8.64e10·61-s + 1.23e11·63-s + 2.74e11·64-s + ⋯ |
| L(s) = 1 | + 0.846·3-s + 2·4-s + 1.96·7-s + 9-s + 1.80·11-s + 1.69·12-s + 3·16-s + 1.26·17-s + 1.66·21-s − 3.12·23-s + 2·25-s + 1.93·27-s + 3.93·28-s − 1.66·29-s + 0.114·31-s + 1.52·33-s + 2·36-s + 1.95·37-s − 3.98·41-s + 3.61·44-s + 2.53·48-s + 49-s + 1.07·51-s + 1.15·59-s − 1.67·61-s + 1.96·63-s + 4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6889 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(18.35612013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.35612013\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 83 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 617 T - 150752 T^{2} - 617 p^{12} T^{3} + p^{24} T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 231577 T + 39786619728 T^{2} - 231577 p^{12} T^{3} + p^{24} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3198553 T + 7092312917088 T^{2} - 3198553 p^{12} T^{3} + p^{24} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 30626737 T + 355374782037408 T^{2} - 30626737 p^{12} T^{3} + p^{24} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 231011422 T + p^{12} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 991523567 T + 629304200710934448 T^{2} + 991523567 p^{12} T^{3} + p^{24} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 101624713 T - 777335201496217392 T^{2} - 101624713 p^{12} T^{3} + p^{24} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5012228257 T + 18539480094429222768 T^{2} - 5012228257 p^{12} T^{3} + p^{24} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9474786718 T + p^{12} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 48671434393 T + \)\(58\!\cdots\!68\)\( T^{2} - 48671434393 p^{12} T^{3} + p^{24} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 86434023647 T + \)\(48\!\cdots\!88\)\( T^{2} + 86434023647 p^{12} T^{3} + p^{24} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
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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
−11.94077177002931132442095274994, −11.81405687740170523787975394119, −11.09308532006320970152662817772, −10.58690422575881913628502002089, −10.04106536208030923888578737051, −9.518712470686704764474983968535, −8.326131608108892350463358363015, −8.321346646336957397823816654304, −7.67908052544686140765495496440, −7.14313871505069713934019061293, −6.54739404053945269896670506587, −6.06462331461534684840056413365, −5.18902716524000367416630308271, −4.45047896146680919095433953157, −3.63434061434975935541261509677, −3.24252135428939675486723024825, −2.21511776253716780369490012603, −1.76134289431743208580118491072, −1.46964608485394452661783844332, −0.973435885860746539521554841406,
0.973435885860746539521554841406, 1.46964608485394452661783844332, 1.76134289431743208580118491072, 2.21511776253716780369490012603, 3.24252135428939675486723024825, 3.63434061434975935541261509677, 4.45047896146680919095433953157, 5.18902716524000367416630308271, 6.06462331461534684840056413365, 6.54739404053945269896670506587, 7.14313871505069713934019061293, 7.67908052544686140765495496440, 8.321346646336957397823816654304, 8.326131608108892350463358363015, 9.518712470686704764474983968535, 10.04106536208030923888578737051, 10.58690422575881913628502002089, 11.09308532006320970152662817772, 11.81405687740170523787975394119, 11.94077177002931132442095274994
