| L(s) = 1 | − 5-s − 2·7-s − 11-s − 13-s − 2·17-s + 8·19-s − 23-s + 5·29-s − 31-s + 2·35-s + 12·37-s − 7·43-s − 7·47-s + 7·49-s − 24·53-s + 55-s + 4·59-s − 10·61-s + 65-s + 4·67-s + 24·71-s + 12·73-s + 2·77-s − 15·79-s − 2·83-s + 2·85-s − 24·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.301·11-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.208·23-s + 0.928·29-s − 0.179·31-s + 0.338·35-s + 1.97·37-s − 1.06·43-s − 1.02·47-s + 49-s − 3.29·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.124·65-s + 0.488·67-s + 2.84·71-s + 1.40·73-s + 0.227·77-s − 1.68·79-s − 0.219·83-s + 0.216·85-s − 2.54·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437506161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437506161\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.811180098204356398035596160162, −8.396147865566301539098797273283, −8.052977308860744675381493226106, −7.69273933037273209090264112468, −7.48206075104480378644952154314, −6.96029358128351868668549425402, −6.46194394731606185759539926355, −6.42973201773822391881252431866, −5.82933769369520018390197761801, −5.40671288368061186665201614622, −4.96802095221989021646959852615, −4.61181274697251050954562419488, −4.25734716646285136907031146503, −3.60704971094001983496791193509, −3.14841587861224694340556817538, −3.05153962687382360164458365677, −2.39830636531264772134802039585, −1.78444074223996047091363462196, −1.06921554210307548667931742432, −0.41774804912269872694536073101,
0.41774804912269872694536073101, 1.06921554210307548667931742432, 1.78444074223996047091363462196, 2.39830636531264772134802039585, 3.05153962687382360164458365677, 3.14841587861224694340556817538, 3.60704971094001983496791193509, 4.25734716646285136907031146503, 4.61181274697251050954562419488, 4.96802095221989021646959852615, 5.40671288368061186665201614622, 5.82933769369520018390197761801, 6.42973201773822391881252431866, 6.46194394731606185759539926355, 6.96029358128351868668549425402, 7.48206075104480378644952154314, 7.69273933037273209090264112468, 8.052977308860744675381493226106, 8.396147865566301539098797273283, 8.811180098204356398035596160162
