| L(s) = 1 | − 270·5-s − 1.11e3·7-s + 5.72e3·11-s + 4.57e3·13-s − 7.31e4·17-s + 1.03e5·19-s − 2.22e4·23-s + 7.81e4·25-s + 1.57e5·29-s + 1.03e5·31-s + 3.00e5·35-s − 1.89e5·37-s − 6.59e5·41-s + 7.57e4·43-s − 4.05e5·47-s + 8.23e5·49-s − 2.69e6·53-s − 1.54e6·55-s + 1.30e6·59-s − 1.83e6·61-s − 1.23e6·65-s − 1.36e6·67-s + 5.42e6·71-s + 5.73e6·73-s − 6.36e6·77-s + 1.12e6·79-s − 5.91e6·83-s + ⋯ |
| L(s) = 1 | − 0.965·5-s − 1.22·7-s + 1.29·11-s + 0.576·13-s − 3.60·17-s + 3.46·19-s − 0.381·23-s + 25-s + 1.19·29-s + 0.626·31-s + 1.18·35-s − 0.615·37-s − 1.49·41-s + 0.145·43-s − 0.569·47-s + 49-s − 2.48·53-s − 1.25·55-s + 0.826·59-s − 1.03·61-s − 0.557·65-s − 0.556·67-s + 1.79·71-s + 1.72·73-s − 1.58·77-s + 0.257·79-s − 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.04873798068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04873798068\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 54 p T - 209 p^{2} T^{2} + 54 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 1112 T + 413001 T^{2} + 1112 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5724 T + 13277005 T^{2} - 5724 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4570 T - 41863617 T^{2} - 4570 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 36558 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 51740 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22248 T - 2909851943 T^{2} + 22248 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 157194 T + 7460077327 T^{2} - 157194 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 103936 T - 16709922015 T^{2} - 103936 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 94834 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 659610 T + 240331078219 T^{2} + 659610 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 75772 T - 266077215123 T^{2} - 75772 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 405648 T - 342072820559 T^{2} + 405648 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1346274 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 1303884 T - 788537999363 T^{2} - 1303884 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 30062 p T + 59127543 p^{2} T^{2} + 30062 p^{8} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1369388 T - 4185488110779 T^{2} + 1369388 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2714040 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2868794 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1129648 T - 17927804382255 T^{2} - 1129648 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 5912028 T + 7816024083157 T^{2} + 5912028 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 897750 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13719074 T + 107414706939363 T^{2} + 13719074 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.96648125326370424167546061746, −10.05951127372580387803498176460, −9.497691210515241476256525800389, −9.328174919981083565654011212845, −8.873749744867254034745447490361, −8.277847231514560599102578288380, −7.919290744061293111750294143666, −6.96574043360384886437482420760, −6.76818002561486737342000569658, −6.61643673653370236252139030534, −5.94950038222735152900788866351, −4.89623282934987244256125614795, −4.83217720888099950701561914413, −3.82890436514951123766563072424, −3.72320177272036356927991141471, −2.99581203637746284474642805173, −2.54430919253930439269969692484, −1.42875630174275681341853466601, −1.06419246943259190529503173783, −0.05410410626730934161571091430,
0.05410410626730934161571091430, 1.06419246943259190529503173783, 1.42875630174275681341853466601, 2.54430919253930439269969692484, 2.99581203637746284474642805173, 3.72320177272036356927991141471, 3.82890436514951123766563072424, 4.83217720888099950701561914413, 4.89623282934987244256125614795, 5.94950038222735152900788866351, 6.61643673653370236252139030534, 6.76818002561486737342000569658, 6.96574043360384886437482420760, 7.919290744061293111750294143666, 8.277847231514560599102578288380, 8.873749744867254034745447490361, 9.328174919981083565654011212845, 9.497691210515241476256525800389, 10.05951127372580387803498176460, 10.96648125326370424167546061746
