| L(s) = 1 | + 3-s − 4-s − 3·7-s + 9-s − 12-s − 4·13-s + 16-s − 4·17-s + 19-s − 3·21-s − 2·23-s + 4·25-s + 27-s + 3·28-s + 4·29-s − 3·31-s − 36-s − 4·39-s − 9·41-s − 3·43-s − 3·47-s + 48-s − 6·49-s − 4·51-s + 4·52-s + 53-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 1.13·7-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.229·19-s − 0.654·21-s − 0.417·23-s + 4/5·25-s + 0.192·27-s + 0.566·28-s + 0.742·29-s − 0.538·31-s − 1/6·36-s − 0.640·39-s − 1.40·41-s − 0.457·43-s − 0.437·47-s + 0.144·48-s − 6/7·49-s − 0.560·51-s + 0.554·52-s + 0.137·53-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 241 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 33 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 54 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 48 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.5207130434, −13.9435972994, −13.7190545096, −13.0981906256, −12.8931571266, −12.4335769544, −11.9848222385, −11.4438105409, −10.8449709061, −10.2023127614, −9.91159290452, −9.58262535203, −8.99458700138, −8.59029052440, −8.15260125735, −7.41667242815, −6.94132174269, −6.56998542045, −5.89916494553, −5.10151325177, −4.64772941105, −4.00824721948, −3.16762705462, −2.81324437929, −1.74774678827, 0,
1.74774678827, 2.81324437929, 3.16762705462, 4.00824721948, 4.64772941105, 5.10151325177, 5.89916494553, 6.56998542045, 6.94132174269, 7.41667242815, 8.15260125735, 8.59029052440, 8.99458700138, 9.58262535203, 9.91159290452, 10.2023127614, 10.8449709061, 11.4438105409, 11.9848222385, 12.4335769544, 12.8931571266, 13.0981906256, 13.7190545096, 13.9435972994, 14.5207130434
