| L(s) = 1 | + 4-s + 6·9-s + 12·11-s + 4·16-s + 24·19-s − 12·31-s + 6·36-s + 12·41-s + 12·44-s − 11·49-s − 18·61-s + 11·64-s + 24·76-s − 32·79-s + 27·81-s − 6·89-s + 72·99-s − 30·101-s − 10·109-s + 62·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2·9-s + 3.61·11-s + 16-s + 5.50·19-s − 2.15·31-s + 36-s + 1.87·41-s + 1.80·44-s − 1.57·49-s − 2.30·61-s + 11/8·64-s + 2.75·76-s − 3.60·79-s + 3·81-s − 0.635·89-s + 7.23·99-s − 2.98·101-s − 0.957·109-s + 5.63·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.968097147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.968097147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 134 T^{2} + 12627 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.54921814427815681258993273403, −7.50428485946865767730072012788, −7.31033368435066735433427625140, −7.29566859442582246722274483474, −6.91949507923541747152764644619, −6.80096621583653811954925738727, −6.39972015410778239529421948402, −6.20141644342548783305907344934, −5.83025680988162103032916167468, −5.68339366934249967285302593489, −5.48920790007260281755087379050, −5.08913806255409155908644584269, −4.70381502499563334361396306879, −4.66737280527091704175577676502, −4.14155564474991405239456537291, −3.81510172024577813546451272901, −3.72082600735245644150677243387, −3.40872774406492731852483830527, −3.35747657652967672257394437758, −2.77490445959488058314570683135, −2.44170852374756167251648637033, −1.43417768136278236277081248175, −1.41238821761972006969970198869, −1.24492861112514253003065757861, −1.23377345390456658824732183758,
1.23377345390456658824732183758, 1.24492861112514253003065757861, 1.41238821761972006969970198869, 1.43417768136278236277081248175, 2.44170852374756167251648637033, 2.77490445959488058314570683135, 3.35747657652967672257394437758, 3.40872774406492731852483830527, 3.72082600735245644150677243387, 3.81510172024577813546451272901, 4.14155564474991405239456537291, 4.66737280527091704175577676502, 4.70381502499563334361396306879, 5.08913806255409155908644584269, 5.48920790007260281755087379050, 5.68339366934249967285302593489, 5.83025680988162103032916167468, 6.20141644342548783305907344934, 6.39972015410778239529421948402, 6.80096621583653811954925738727, 6.91949507923541747152764644619, 7.29566859442582246722274483474, 7.31033368435066735433427625140, 7.50428485946865767730072012788, 7.54921814427815681258993273403
