| L(s) = 1 | − 4-s + 2·7-s − 2·13-s + 16-s + 2·19-s + 25-s − 2·28-s + 3·49-s + 2·52-s − 2·61-s − 64-s − 2·76-s − 2·79-s − 4·91-s − 2·97-s − 100-s − 103-s + 2·112-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s − 2·13-s + 16-s + 2·19-s + 25-s − 2·28-s + 3·49-s + 2·52-s − 2·61-s − 64-s − 2·76-s − 2·79-s − 4·91-s − 2·97-s − 100-s − 103-s + 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9641756728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9641756728\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14964807652696624475748911975, −9.424530653515854170670348671416, −8.597951751018079694928303347874, −7.70349933383748826675170633138, −7.31500717223069122445846024011, −5.45090338996913290308917459919, −4.99955387095898289131834956924, −4.35908738223496884988788637839, −2.83672835303696180436302962013, −1.35173023458489428413810758253,
1.35173023458489428413810758253, 2.83672835303696180436302962013, 4.35908738223496884988788637839, 4.99955387095898289131834956924, 5.45090338996913290308917459919, 7.31500717223069122445846024011, 7.70349933383748826675170633138, 8.597951751018079694928303347874, 9.424530653515854170670348671416, 10.14964807652696624475748911975
