| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 2·9-s + 2·11-s + 2·13-s + 4·15-s − 4·17-s − 10·19-s + 4·21-s − 8·23-s − 2·25-s − 6·27-s − 4·31-s − 4·33-s + 4·35-s + 4·37-s − 4·39-s + 4·41-s − 12·43-s − 4·45-s + 4·47-s + 3·49-s + 8·51-s − 8·53-s − 4·55-s + 20·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.970·17-s − 2.29·19-s + 0.872·21-s − 1.66·23-s − 2/5·25-s − 1.15·27-s − 0.718·31-s − 0.696·33-s + 0.676·35-s + 0.657·37-s − 0.640·39-s + 0.624·41-s − 1.82·43-s − 0.596·45-s + 0.583·47-s + 3/7·49-s + 1.12·51-s − 1.09·53-s − 0.539·55-s + 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24285184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24285184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.188950646751120878691221061581, −7.61944233186616319261604639665, −7.19984593741056834631719558655, −7.08165178092361835927635667885, −6.34999092035000169199541245202, −6.26745258021015195367252186597, −5.94691766288914587101927752645, −5.92884457159968242053654636700, −5.00504550951794428165403767491, −4.69054128305305828139642868638, −4.43410321497715604673580060442, −3.75472585477969035864056160377, −3.72843140864384130657532443310, −3.50952630587609947277776674778, −2.27935037462662526764598514628, −2.27567721136509897034684087783, −1.69584577146011347095591851993, −0.793546722066651161761575576108, 0, 0,
0.793546722066651161761575576108, 1.69584577146011347095591851993, 2.27567721136509897034684087783, 2.27935037462662526764598514628, 3.50952630587609947277776674778, 3.72843140864384130657532443310, 3.75472585477969035864056160377, 4.43410321497715604673580060442, 4.69054128305305828139642868638, 5.00504550951794428165403767491, 5.92884457159968242053654636700, 5.94691766288914587101927752645, 6.26745258021015195367252186597, 6.34999092035000169199541245202, 7.08165178092361835927635667885, 7.19984593741056834631719558655, 7.61944233186616319261604639665, 8.188950646751120878691221061581
