| L(s) = 1 | + 4-s + 8·5-s + 9-s + 2·11-s − 2·19-s + 8·20-s + 38·25-s + 20·29-s − 16·31-s + 36-s + 12·41-s + 2·44-s + 8·45-s − 13·49-s + 16·55-s − 16·59-s + 16·61-s − 64-s + 8·71-s − 2·76-s − 40·79-s + 6·89-s − 16·95-s + 2·99-s + 38·100-s − 20·101-s + 24·109-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.57·5-s + 1/3·9-s + 0.603·11-s − 0.458·19-s + 1.78·20-s + 38/5·25-s + 3.71·29-s − 2.87·31-s + 1/6·36-s + 1.87·41-s + 0.301·44-s + 1.19·45-s − 1.85·49-s + 2.15·55-s − 2.08·59-s + 2.04·61-s − 1/8·64-s + 0.949·71-s − 0.229·76-s − 4.50·79-s + 0.635·89-s − 1.64·95-s + 0.201·99-s + 19/5·100-s − 1.99·101-s + 2.29·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.535479830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.535479830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.560174759770139121553233141623, −7.77002169395514653493916969101, −7.68935433415672300013092733075, −7.44786594943298021893363900112, −7.09648917292950132534563381989, −6.66041767807158524897672776051, −6.50371635261990344327958270961, −6.44007813862315983233171382307, −6.42196099611231895488667327820, −5.80889053219484239571850353296, −5.80197904663898141899208587181, −5.44696032989317189387298553272, −5.16445054282806340576777775741, −5.00627161008139950798055829389, −4.60204801114886725675708136275, −4.18107460597994745869538013381, −4.16516551842438787693894217636, −3.27805411638405527022160347675, −3.18655276279720481297731573965, −2.71976199884169684737311923875, −2.35356654591671720579955790969, −2.30426358543501402544020275408, −1.59258948705493560202405968137, −1.50032075637314203159365632909, −1.08256728239925385301233898069,
1.08256728239925385301233898069, 1.50032075637314203159365632909, 1.59258948705493560202405968137, 2.30426358543501402544020275408, 2.35356654591671720579955790969, 2.71976199884169684737311923875, 3.18655276279720481297731573965, 3.27805411638405527022160347675, 4.16516551842438787693894217636, 4.18107460597994745869538013381, 4.60204801114886725675708136275, 5.00627161008139950798055829389, 5.16445054282806340576777775741, 5.44696032989317189387298553272, 5.80197904663898141899208587181, 5.80889053219484239571850353296, 6.42196099611231895488667327820, 6.44007813862315983233171382307, 6.50371635261990344327958270961, 6.66041767807158524897672776051, 7.09648917292950132534563381989, 7.44786594943298021893363900112, 7.68935433415672300013092733075, 7.77002169395514653493916969101, 8.560174759770139121553233141623
