| L(s) = 1 | + 6·2-s + 17·4-s + 30·8-s − 12·9-s + 40·16-s − 72·18-s − 8·23-s + 3·25-s − 2·29-s + 54·32-s − 204·36-s + 6·37-s − 12·43-s − 48·46-s + 18·50-s − 10·53-s − 12·58-s + 79·64-s − 36·71-s − 360·72-s + 36·74-s + 12·79-s + 90·81-s − 72·86-s − 136·92-s + 51·100-s − 60·106-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 17/2·4-s + 10.6·8-s − 4·9-s + 10·16-s − 16.9·18-s − 1.66·23-s + 3/5·25-s − 0.371·29-s + 9.54·32-s − 34·36-s + 0.986·37-s − 1.82·43-s − 7.07·46-s + 2.54·50-s − 1.37·53-s − 1.57·58-s + 79/8·64-s − 4.27·71-s − 42.4·72-s + 4.18·74-s + 1.35·79-s + 10·81-s − 7.76·86-s − 14.1·92-s + 5.09·100-s − 5.82·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.148519332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.148519332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 - 3 T^{2} - 16 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 5 T^{2} - 264 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 69 T^{2} + 3080 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 42 T^{2} - 445 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 66 T^{2} + 875 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 126 T^{2} + 7955 T^{4} + 126 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 142 T^{2} + 10755 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.69105833834163066147662545448, −7.33063814211889802054274453764, −6.63191143533260663629911302754, −6.54672213608285406676971517527, −6.47042354969362534499273867788, −6.20517730619048813773823513123, −5.87821654605124376207972268273, −5.84518494028935359643113665102, −5.50841329425091099436613967523, −5.28934887138452257283643879287, −5.23068533018641320486376213984, −5.15606683487691802947533179462, −4.50105566736686953522854337611, −4.42400311598368395792882317043, −4.39748896659390997246022065177, −3.76588395505877463212563529608, −3.60263426001996451256452542916, −3.51414917348911455170482919148, −3.01153593017361447647960157986, −2.96665323100623014057856891250, −2.63937966139072992820350387418, −2.55833679511662970504883023048, −1.87121257033801735848186767441, −1.45624701312338406379266520779, −0.27538367513006075059683810003,
0.27538367513006075059683810003, 1.45624701312338406379266520779, 1.87121257033801735848186767441, 2.55833679511662970504883023048, 2.63937966139072992820350387418, 2.96665323100623014057856891250, 3.01153593017361447647960157986, 3.51414917348911455170482919148, 3.60263426001996451256452542916, 3.76588395505877463212563529608, 4.39748896659390997246022065177, 4.42400311598368395792882317043, 4.50105566736686953522854337611, 5.15606683487691802947533179462, 5.23068533018641320486376213984, 5.28934887138452257283643879287, 5.50841329425091099436613967523, 5.84518494028935359643113665102, 5.87821654605124376207972268273, 6.20517730619048813773823513123, 6.47042354969362534499273867788, 6.54672213608285406676971517527, 6.63191143533260663629911302754, 7.33063814211889802054274453764, 7.69105833834163066147662545448
