| L(s) = 1 | − 2.23·3-s − 5-s − 2.23·7-s + 2.00·9-s + 4.47·11-s + 13-s + 2.23·15-s − 7·17-s + 5.00·21-s + 4.47·23-s − 4·25-s + 2.23·27-s − 6·29-s + 8.94·31-s − 10.0·33-s + 2.23·35-s + 3·37-s − 2.23·39-s + 2.23·43-s − 2.00·45-s − 2.23·47-s − 1.99·49-s + 15.6·51-s + 14·53-s − 4.47·55-s + 8.94·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 0.447·5-s − 0.845·7-s + 0.666·9-s + 1.34·11-s + 0.277·13-s + 0.577·15-s − 1.69·17-s + 1.09·21-s + 0.932·23-s − 0.800·25-s + 0.430·27-s − 1.11·29-s + 1.60·31-s − 1.74·33-s + 0.377·35-s + 0.493·37-s − 0.358·39-s + 0.340·43-s − 0.298·45-s − 0.326·47-s − 0.285·49-s + 2.19·51-s + 1.92·53-s − 0.603·55-s + 1.16·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 6.70T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.356446329897969968906192884305, −7.13503291908611495675875345264, −6.62186274278520178045814869482, −6.17214968752126037698886333716, −5.30420051346238846122644636105, −4.31162401500747686236355179885, −3.80952725971069540845244974377, −2.54920858987384065991528285586, −1.09883388111917475906497954617, 0,
1.09883388111917475906497954617, 2.54920858987384065991528285586, 3.80952725971069540845244974377, 4.31162401500747686236355179885, 5.30420051346238846122644636105, 6.17214968752126037698886333716, 6.62186274278520178045814869482, 7.13503291908611495675875345264, 8.356446329897969968906192884305
