| L(s) = 1 | − 3-s + 5-s + 2·11-s + 2·13-s − 15-s − 7·17-s + 5·19-s + 25-s + 4·27-s − 10·29-s − 5·31-s − 2·33-s − 10·37-s − 2·39-s + 2·41-s − 3·43-s − 8·47-s − 10·49-s + 7·51-s − 5·53-s + 2·55-s − 5·57-s − 2·59-s − 4·61-s + 2·65-s − 13·67-s + 71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.603·11-s + 0.554·13-s − 0.258·15-s − 1.69·17-s + 1.14·19-s + 1/5·25-s + 0.769·27-s − 1.85·29-s − 0.898·31-s − 0.348·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.457·43-s − 1.16·47-s − 1.42·49-s + 0.980·51-s − 0.686·53-s + 0.269·55-s − 0.662·57-s − 0.260·59-s − 0.512·61-s + 0.248·65-s − 1.58·67-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 331 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 130 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 230 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T - 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−13.4962167111, −13.0203538992, −12.5856475638, −12.1414143376, −11.7340518940, −11.1407354285, −10.9778673120, −10.7762072281, −9.93204844465, −9.56785994281, −9.20532264973, −8.80731187151, −8.30909187397, −7.72779837778, −7.10968471204, −6.74220283866, −6.34616686967, −5.79968090222, −5.28410697971, −4.86429399519, −4.21779112833, −3.52562832638, −3.06054333309, −1.94532391276, −1.50368718840, 0,
1.50368718840, 1.94532391276, 3.06054333309, 3.52562832638, 4.21779112833, 4.86429399519, 5.28410697971, 5.79968090222, 6.34616686967, 6.74220283866, 7.10968471204, 7.72779837778, 8.30909187397, 8.80731187151, 9.20532264973, 9.56785994281, 9.93204844465, 10.7762072281, 10.9778673120, 11.1407354285, 11.7340518940, 12.1414143376, 12.5856475638, 13.0203538992, 13.4962167111
