| L(s) = 1 | − 1.18e3·9-s − 9.08e3·11-s + 7.75e4·19-s + 1.44e5·29-s − 6.13e5·31-s + 5.28e5·41-s + 2.81e6·49-s − 2.24e6·59-s + 4.51e6·61-s − 1.24e6·71-s + 8.66e6·79-s − 6.57e6·81-s − 1.20e7·89-s + 1.07e7·99-s + 1.54e6·101-s + 2.08e7·109-s − 2.00e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.88e8·169-s + ⋯ |
| L(s) = 1 | − 0.539·9-s − 2.05·11-s + 2.59·19-s + 1.10·29-s − 3.69·31-s + 1.19·41-s + 3.41·49-s − 1.42·59-s + 2.54·61-s − 0.412·71-s + 1.97·79-s − 1.37·81-s − 1.81·89-s + 1.11·99-s + 0.149·101-s + 1.54·109-s − 0.103·121-s + 3.00·169-s − 1.39·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.08644563621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08644563621\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 1180 T^{2} + 885382 p^{2} T^{4} + 1180 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2812500 T^{2} + 3331601848198 T^{4} - 2812500 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4544 T + 31976326 T^{2} + 4544 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 188539340 T^{2} + 16409454868245078 T^{4} - 188539340 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1054534780 T^{2} + 535129713917980358 T^{4} - 1054534780 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2040 p T + 113449762 p T^{2} - 2040 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5888629460 T^{2} + 31659696543257618118 T^{4} - 5888629460 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 72260 T + 6846819118 T^{2} - 72260 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 306824 T + 77964629966 T^{2} + 306824 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 273043279340 T^{2} + \)\(35\!\cdots\!78\)\( T^{4} - 273043279340 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 264364 T + 161786388886 T^{2} - 264364 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 714119572100 T^{2} + \)\(24\!\cdots\!98\)\( T^{4} - 714119572100 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1706308901620 T^{2} + \)\(12\!\cdots\!38\)\( T^{4} - 1706308901620 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1405450112620 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - 1405450112620 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 1120120 T + 1362334883638 T^{2} + 1120120 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 13085764398180 T^{2} + \)\(10\!\cdots\!58\)\( T^{4} - 13085764398180 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 621784 T + 17914494152446 T^{2} + 621784 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 25973590426460 T^{2} + \)\(33\!\cdots\!18\)\( T^{4} - 25973590426460 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24038967921380 T^{2} - \)\(13\!\cdots\!42\)\( T^{4} - 24038967921380 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6025620 T + 89865866149558 T^{2} + 6025620 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 264612844151420 T^{2} + \)\(30\!\cdots\!38\)\( T^{4} - 264612844151420 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−7.09433987590546021215932475976, −6.78560662261644553489780872780, −6.49493214538967646455195560314, −6.13545309737095675133613228747, −5.61723067415261720381248308101, −5.60277421073355091231339236556, −5.39033662090577179279533612676, −5.30820399879821900283407558092, −5.30080765936544268661563653095, −4.54908149208474152723578358097, −4.47086558203880405403875890564, −3.97835859883630166132138115744, −3.92135797192464729046315941153, −3.42654744335063826945706093056, −3.17460544010588204950825370308, −3.00523475630506274519656812089, −2.60902907615621397489397615859, −2.53723649005044181580170518909, −2.05323557351448525339766254287, −1.85159152900088429812577339675, −1.49439440652457028379259421029, −0.898225616622116224618004423504, −0.811962534432449833931098993781, −0.59226014462963066997149044299, −0.03206930814148892923898344113,
0.03206930814148892923898344113, 0.59226014462963066997149044299, 0.811962534432449833931098993781, 0.898225616622116224618004423504, 1.49439440652457028379259421029, 1.85159152900088429812577339675, 2.05323557351448525339766254287, 2.53723649005044181580170518909, 2.60902907615621397489397615859, 3.00523475630506274519656812089, 3.17460544010588204950825370308, 3.42654744335063826945706093056, 3.92135797192464729046315941153, 3.97835859883630166132138115744, 4.47086558203880405403875890564, 4.54908149208474152723578358097, 5.30080765936544268661563653095, 5.30820399879821900283407558092, 5.39033662090577179279533612676, 5.60277421073355091231339236556, 5.61723067415261720381248308101, 6.13545309737095675133613228747, 6.49493214538967646455195560314, 6.78560662261644553489780872780, 7.09433987590546021215932475976
