| L(s) = 1 | + 4·4-s − 6·9-s + 24·11-s + 12·16-s − 48·23-s + 28·25-s − 24·36-s − 44·37-s + 28·43-s + 96·44-s − 240·53-s + 32·64-s − 220·67-s + 312·71-s + 20·79-s + 27·81-s − 192·92-s − 144·99-s + 112·100-s + 576·107-s + 580·109-s + 168·113-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s − 2/3·9-s + 2.18·11-s + 3/4·16-s − 2.08·23-s + 1.11·25-s − 2/3·36-s − 1.18·37-s + 0.651·43-s + 2.18·44-s − 4.52·53-s + 1/2·64-s − 3.28·67-s + 4.39·71-s + 0.253·79-s + 1/3·81-s − 2.08·92-s − 1.45·99-s + 1.11·100-s + 5.38·107-s + 5.32·109-s + 1.48·113-s − 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.261035945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.261035945\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 294 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 724 T^{2} + 279654 T^{4} - 724 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 58 p T^{2} + 550131 T^{4} - 58 p^{5} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 24 T + 554 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1678 T^{2} + 1875 p^{2} T^{4} - 1678 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 22 T + 2571 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5884 T^{2} + 18275334 T^{4} - 5884 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 120 T + 8570 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11476 T^{2} + 56933574 T^{4} - 11476 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13252 T^{2} + 70932006 T^{4} - 13252 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 110 T + 8475 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 156 T + 178 p T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1390 T^{2} + 43340307 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 3795 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 15410 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−8.266277398457491535879011497439, −8.240609907399805199154632531170, −7.68142968568505687874310666041, −7.57950531500514982464906056569, −7.31630086545925104361073685134, −7.03143213549751254061718245453, −6.72340559218184047097906368331, −6.37149709895808899239751472455, −6.19875004119198641798718502491, −6.05124822937860812781147114308, −6.04656624810864961442251854213, −5.42628072548673656266152104409, −5.11282888226602967638897618609, −4.68397621822193774538568887843, −4.44891346847749213634823803717, −4.32543592968467614998065266917, −3.62629660272084183706746554365, −3.45067551127870137981916210702, −3.13634997340052557505366147383, −3.10067168447903387318361454303, −2.06351474936104447191572524978, −1.98419262950485146374760828388, −1.81273904851555946954955418094, −1.04649667383276184100995630739, −0.51122406546412885855857359180,
0.51122406546412885855857359180, 1.04649667383276184100995630739, 1.81273904851555946954955418094, 1.98419262950485146374760828388, 2.06351474936104447191572524978, 3.10067168447903387318361454303, 3.13634997340052557505366147383, 3.45067551127870137981916210702, 3.62629660272084183706746554365, 4.32543592968467614998065266917, 4.44891346847749213634823803717, 4.68397621822193774538568887843, 5.11282888226602967638897618609, 5.42628072548673656266152104409, 6.04656624810864961442251854213, 6.05124822937860812781147114308, 6.19875004119198641798718502491, 6.37149709895808899239751472455, 6.72340559218184047097906368331, 7.03143213549751254061718245453, 7.31630086545925104361073685134, 7.57950531500514982464906056569, 7.68142968568505687874310666041, 8.240609907399805199154632531170, 8.266277398457491535879011497439
