| L(s) = 1 | + 8·2-s − 114·5-s + 1.57e3·7-s − 512·8-s − 912·10-s + 7.33e3·11-s + 3.80e3·13-s + 1.26e4·14-s − 4.09e3·16-s + 1.32e4·17-s + 4.97e4·19-s + 5.86e4·22-s + 4.14e4·23-s + 7.81e4·25-s + 3.04e4·26-s − 4.16e4·29-s − 3.31e4·31-s + 1.05e5·34-s − 1.79e5·35-s − 7.29e4·37-s + 3.97e5·38-s + 5.83e4·40-s − 6.39e5·41-s + 1.56e5·43-s + 3.31e5·46-s − 4.33e5·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.407·5-s + 1.73·7-s − 0.353·8-s − 0.288·10-s + 1.66·11-s + 0.479·13-s + 1.22·14-s − 1/4·16-s + 0.652·17-s + 1.66·19-s + 1.17·22-s + 0.710·23-s + 25-s + 0.339·26-s − 0.316·29-s − 0.199·31-s + 0.461·34-s − 0.708·35-s − 0.236·37-s + 1.17·38-s + 0.144·40-s − 1.44·41-s + 0.300·43-s + 0.502·46-s − 0.609·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.145875895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.145875895\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 114 T - 65129 T^{2} + 114 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1576 T + 1660233 T^{2} - 1576 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7332 T + 34271053 T^{2} - 7332 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3802 T - 48293313 T^{2} - 3802 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6606 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 24860 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 41448 T - 1686888743 T^{2} - 41448 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 41610 T - 15518484209 T^{2} + 41610 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 33152 T - 26413559007 T^{2} + 33152 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 36466 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 639078 T + 213666416203 T^{2} + 639078 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 156412 T - 247353897363 T^{2} - 156412 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 433776 T - 318461502287 T^{2} + 433776 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 786078 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 745140 T - 1933417865219 T^{2} - 745140 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1660618 T - 385090694097 T^{2} - 1660618 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3290836 T + 4768889973573 T^{2} - 3290836 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5716152 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2659898 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 3807440 T - 4707309632559 T^{2} + 3807440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2229468 T - 22165523426603 T^{2} - 2229468 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5991210 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 4060126 T - 64313661342237 T^{2} - 4060126 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.05536288576693760670506925981, −11.32516085235412545825001353390, −11.23066944931637494197649856409, −10.39784594693959589143016274803, −9.749390618610296677921067769477, −9.109919327656964133616508052102, −8.785324837419454948526033498790, −8.077556745212407564072341138245, −7.77020222569872154160653424785, −6.85909417166838016699706226707, −6.71927974517994031950015525191, −5.49639370276721894243733052651, −5.41405277950669365474125001517, −4.61208532147077761529310886004, −4.21337700194390129404539807148, −3.40778009914007102565989108958, −3.05107490869168294742090891839, −1.65115856609952046296640652370, −1.39564175196046905244676515181, −0.69269984240550148895587353163,
0.69269984240550148895587353163, 1.39564175196046905244676515181, 1.65115856609952046296640652370, 3.05107490869168294742090891839, 3.40778009914007102565989108958, 4.21337700194390129404539807148, 4.61208532147077761529310886004, 5.41405277950669365474125001517, 5.49639370276721894243733052651, 6.71927974517994031950015525191, 6.85909417166838016699706226707, 7.77020222569872154160653424785, 8.077556745212407564072341138245, 8.785324837419454948526033498790, 9.109919327656964133616508052102, 9.749390618610296677921067769477, 10.39784594693959589143016274803, 11.23066944931637494197649856409, 11.32516085235412545825001353390, 12.05536288576693760670506925981
