| L(s) = 1 | − 2-s − 4·7-s + 8-s + 4·13-s + 4·14-s − 16-s − 6·17-s − 2·19-s − 6·23-s − 10·25-s − 4·26-s + 6·29-s − 5·31-s + 6·34-s − 8·37-s + 2·38-s − 3·41-s − 2·43-s + 6·46-s + 3·47-s + 9·49-s + 10·50-s − 6·53-s − 4·56-s − 6·58-s − 12·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.51·7-s + 0.353·8-s + 1.10·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.458·19-s − 1.25·23-s − 2·25-s − 0.784·26-s + 1.11·29-s − 0.898·31-s + 1.02·34-s − 1.31·37-s + 0.324·38-s − 0.468·41-s − 0.304·43-s + 0.884·46-s + 0.437·47-s + 9/7·49-s + 1.41·50-s − 0.824·53-s − 0.534·56-s − 0.787·58-s − 1.56·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−9.415507279830890488392113325475, −9.321625853821069835027506088770, −8.704513245761455668575784358552, −8.671512861062864257852877442902, −7.984718837291666890931885437457, −7.69427999619948725340724044947, −6.97326153982318964600085723884, −6.85029001526945372571471571174, −6.08853665698931840005713745024, −6.04194366736133261960039293027, −5.66320204799691869378309545078, −4.64545205436501340054512287153, −4.30772461514516820171622143200, −3.89731672134636019352475857981, −3.25213847503822875344463596991, −2.87457999811761605231113501978, −1.88626956593863709125787110557, −1.60666715479128206862102868690, 0, 0,
1.60666715479128206862102868690, 1.88626956593863709125787110557, 2.87457999811761605231113501978, 3.25213847503822875344463596991, 3.89731672134636019352475857981, 4.30772461514516820171622143200, 4.64545205436501340054512287153, 5.66320204799691869378309545078, 6.04194366736133261960039293027, 6.08853665698931840005713745024, 6.85029001526945372571471571174, 6.97326153982318964600085723884, 7.69427999619948725340724044947, 7.984718837291666890931885437457, 8.671512861062864257852877442902, 8.704513245761455668575784358552, 9.321625853821069835027506088770, 9.415507279830890488392113325475
