| L(s) = 1 | + 2-s + 5-s − 7-s + 8-s − 2·9-s + 10-s − 3·11-s + 3·13-s − 14-s − 16-s + 7·17-s − 2·18-s − 3·19-s − 3·22-s − 23-s − 4·25-s + 3·26-s − 3·27-s − 6·29-s − 3·31-s − 6·32-s + 7·34-s − 35-s + 9·37-s − 3·38-s + 40-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s + 0.832·13-s − 0.267·14-s − 1/4·16-s + 1.69·17-s − 0.471·18-s − 0.688·19-s − 0.639·22-s − 0.208·23-s − 4/5·25-s + 0.588·26-s − 0.577·27-s − 1.11·29-s − 0.538·31-s − 1.06·32-s + 1.20·34-s − 0.169·35-s + 1.47·37-s − 0.486·38-s + 0.158·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5865 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5865 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162336540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162336540\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−17.1278154275, −16.7028725153, −16.3598779481, −15.7453914252, −15.1497621867, −14.6605492541, −14.1295388742, −13.6115424663, −13.1773856155, −12.9392459008, −12.1354170234, −11.5390713772, −11.0074448089, −10.3173417782, −9.91305206370, −9.13814223341, −8.52957943630, −7.74916711982, −7.28515800309, −6.13836542611, −5.71707465741, −5.18747621405, −4.06676193965, −3.39608649967, −2.15850873965,
2.15850873965, 3.39608649967, 4.06676193965, 5.18747621405, 5.71707465741, 6.13836542611, 7.28515800309, 7.74916711982, 8.52957943630, 9.13814223341, 9.91305206370, 10.3173417782, 11.0074448089, 11.5390713772, 12.1354170234, 12.9392459008, 13.1773856155, 13.6115424663, 14.1295388742, 14.6605492541, 15.1497621867, 15.7453914252, 16.3598779481, 16.7028725153, 17.1278154275
