| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 8-s − 10-s + 15-s − 16-s + 3·17-s − 3·19-s + 4·23-s + 24-s − 27-s − 30-s + 2·31-s − 3·34-s + 3·38-s + 40-s − 4·46-s − 48-s + 49-s + 3·51-s + 54-s − 3·57-s − 2·62-s + 64-s + 4·69-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 8-s − 10-s + 15-s − 16-s + 3·17-s − 3·19-s + 4·23-s + 24-s − 27-s − 30-s + 2·31-s − 3·34-s + 3·38-s + 40-s − 4·46-s − 48-s + 49-s + 3·51-s + 54-s − 3·57-s − 2·62-s + 64-s + 4·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.262248942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262248942\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.336428078378838379738944091336, −9.249814747813722058007564429935, −8.886868181183735083465991215576, −8.558966741309767538631948334439, −8.178602701647745317715837367377, −7.84420923158447677740126920010, −7.48538907443816061816943423480, −6.99454773990485624562995354775, −6.41724347292239628576435581927, −6.32352914776660542738198691090, −5.47931182145122082950935467777, −5.21156969868283959431064851265, −4.88976189809440580387810407107, −4.09726978274268579648477842397, −3.82643254176750634754839595636, −2.99440582312800100788105855446, −2.73349509836783936516703775278, −2.30538951034803661360558472117, −1.27597742093828211389884274589, −1.20760612367893974666474132963,
1.20760612367893974666474132963, 1.27597742093828211389884274589, 2.30538951034803661360558472117, 2.73349509836783936516703775278, 2.99440582312800100788105855446, 3.82643254176750634754839595636, 4.09726978274268579648477842397, 4.88976189809440580387810407107, 5.21156969868283959431064851265, 5.47931182145122082950935467777, 6.32352914776660542738198691090, 6.41724347292239628576435581927, 6.99454773990485624562995354775, 7.48538907443816061816943423480, 7.84420923158447677740126920010, 8.178602701647745317715837367377, 8.558966741309767538631948334439, 8.886868181183735083465991215576, 9.249814747813722058007564429935, 9.336428078378838379738944091336
