| L(s) = 1 | − 2·3-s − 3.31·5-s + 3.31·7-s + 9-s + 5·11-s + 6.63·15-s + 5·17-s + 19-s − 6.63·21-s + 6.63·23-s + 6·25-s + 4·27-s − 6.63·29-s − 10·33-s − 11·35-s + 6.63·37-s + 6·41-s − 43-s − 3.31·45-s − 9.94·47-s + 4·49-s − 10·51-s + 13.2·53-s − 16.5·55-s − 2·57-s − 6·59-s − 9.94·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.48·5-s + 1.25·7-s + 0.333·9-s + 1.50·11-s + 1.71·15-s + 1.21·17-s + 0.229·19-s − 1.44·21-s + 1.38·23-s + 1.20·25-s + 0.769·27-s − 1.23·29-s − 1.74·33-s − 1.85·35-s + 1.09·37-s + 0.937·41-s − 0.152·43-s − 0.494·45-s − 1.45·47-s + 0.571·49-s − 1.40·51-s + 1.82·53-s − 2.23·55-s − 0.264·57-s − 0.781·59-s − 1.27·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.247649593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247649593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 29 | \( 1 + 6.63T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 9.94T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.067205164676451621213300957554, −7.56091010996316573565695497531, −6.90242670636169450566330297932, −6.03555115459770356723542639384, −5.25762982876777096231523374680, −4.60611409192651308942943323164, −3.95544101904848369680286539119, −3.12131019385518060962220064642, −1.45119405589658494255691524224, −0.73517404529325591849329543340,
0.73517404529325591849329543340, 1.45119405589658494255691524224, 3.12131019385518060962220064642, 3.95544101904848369680286539119, 4.60611409192651308942943323164, 5.25762982876777096231523374680, 6.03555115459770356723542639384, 6.90242670636169450566330297932, 7.56091010996316573565695497531, 8.067205164676451621213300957554
