| L(s) = 1 | − 2·3-s + 2·4-s − 3·5-s + 3·9-s − 4·12-s − 2·13-s + 6·15-s − 6·17-s − 7·19-s − 6·20-s − 3·23-s + 5·25-s − 10·27-s − 18·29-s + 5·31-s + 6·36-s − 2·37-s + 4·39-s + 12·41-s − 2·43-s − 9·45-s + 3·47-s + 12·51-s − 4·52-s + 9·53-s + 14·57-s + 12·60-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s − 1.34·5-s + 9-s − 1.15·12-s − 0.554·13-s + 1.54·15-s − 1.45·17-s − 1.60·19-s − 1.34·20-s − 0.625·23-s + 25-s − 1.92·27-s − 3.34·29-s + 0.898·31-s + 36-s − 0.328·37-s + 0.640·39-s + 1.87·41-s − 0.304·43-s − 1.34·45-s + 0.437·47-s + 1.68·51-s − 0.554·52-s + 1.23·53-s + 1.85·57-s + 1.54·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3005258062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3005258062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−11.21557192654656115929059562471, −10.42152116838089024408421527126, −10.36775331907498145852967320344, −9.440440499179382596042775370057, −9.014215020491032074364170034349, −8.777370686036443083534922794693, −7.73643645509670114451589378129, −7.65027749308717644137194187323, −7.38537725741916127515420207826, −6.70033691910657328299274265644, −6.42059031543428918413001797613, −5.87323786425657077886575995128, −5.51341319935078408308514775901, −4.63903654309338344185590532848, −4.14531066632818420635662892051, −4.09765708120043175992488949210, −3.14799805443533755863361648456, −2.07921993023177662744445569310, −1.99815007761079635100907928774, −0.29757902661670769068819707205,
0.29757902661670769068819707205, 1.99815007761079635100907928774, 2.07921993023177662744445569310, 3.14799805443533755863361648456, 4.09765708120043175992488949210, 4.14531066632818420635662892051, 4.63903654309338344185590532848, 5.51341319935078408308514775901, 5.87323786425657077886575995128, 6.42059031543428918413001797613, 6.70033691910657328299274265644, 7.38537725741916127515420207826, 7.65027749308717644137194187323, 7.73643645509670114451589378129, 8.777370686036443083534922794693, 9.014215020491032074364170034349, 9.440440499179382596042775370057, 10.36775331907498145852967320344, 10.42152116838089024408421527126, 11.21557192654656115929059562471
