| L(s) = 1 | − 2-s + 6·5-s + 8-s − 6·10-s + 12·11-s − 2·13-s − 16-s + 6·17-s + 7·19-s − 12·22-s − 6·23-s + 17·25-s + 2·26-s + 6·29-s − 2·31-s − 6·34-s − 2·37-s − 7·38-s + 6·40-s − 2·43-s + 6·46-s − 17·50-s + 6·53-s + 72·55-s − 6·58-s − 5·61-s + 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.68·5-s + 0.353·8-s − 1.89·10-s + 3.61·11-s − 0.554·13-s − 1/4·16-s + 1.45·17-s + 1.60·19-s − 2.55·22-s − 1.25·23-s + 17/5·25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s − 1.02·34-s − 0.328·37-s − 1.13·38-s + 0.948·40-s − 0.304·43-s + 0.884·46-s − 2.40·50-s + 0.824·53-s + 9.70·55-s − 0.787·58-s − 0.640·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.049478911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.049478911\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.119233135315383801044587941028, −9.041264809231333164630450434066, −8.281190965410225456229977912237, −8.231977883705688061921164225430, −7.23491563636052255265343045813, −7.23182930292208982988464314439, −6.72192966636031351955942389074, −6.34706128620273844702669308203, −5.88026823915732010319848126911, −5.80955960064226799843100178378, −5.40982439305674428384499821208, −4.75495137477734491534799863675, −4.33440100506565243723024952761, −3.80443684959458613798296070366, −3.29709314339897220783545263440, −2.85339256170297661973759243349, −1.90944253051459574682461437155, −1.81903618423166631789326428822, −1.16858505756055753514953615408, −1.04541310225344275330343550942,
1.04541310225344275330343550942, 1.16858505756055753514953615408, 1.81903618423166631789326428822, 1.90944253051459574682461437155, 2.85339256170297661973759243349, 3.29709314339897220783545263440, 3.80443684959458613798296070366, 4.33440100506565243723024952761, 4.75495137477734491534799863675, 5.40982439305674428384499821208, 5.80955960064226799843100178378, 5.88026823915732010319848126911, 6.34706128620273844702669308203, 6.72192966636031351955942389074, 7.23182930292208982988464314439, 7.23491563636052255265343045813, 8.231977883705688061921164225430, 8.281190965410225456229977912237, 9.041264809231333164630450434066, 9.119233135315383801044587941028
