| L(s) = 1 | − 2.18·3-s + 5-s + 1.76·9-s − 3.59·11-s + 5.85·13-s − 2.18·15-s − 6.36·17-s − 4.50·19-s − 2.62·23-s + 25-s + 2.68·27-s + 2.05·29-s + 6.62·31-s + 7.85·33-s − 5.91·37-s − 12.7·39-s + 7.22·41-s + 5.34·43-s + 1.76·45-s − 2.31·47-s + 13.8·51-s − 4.10·53-s − 3.59·55-s + 9.83·57-s + 5.07·59-s − 7.37·61-s + 5.85·65-s + ⋯ |
| L(s) = 1 | − 1.26·3-s + 0.447·5-s + 0.589·9-s − 1.08·11-s + 1.62·13-s − 0.563·15-s − 1.54·17-s − 1.03·19-s − 0.548·23-s + 0.200·25-s + 0.517·27-s + 0.382·29-s + 1.19·31-s + 1.36·33-s − 0.972·37-s − 2.04·39-s + 1.12·41-s + 0.815·43-s + 0.263·45-s − 0.338·47-s + 1.94·51-s − 0.564·53-s − 0.485·55-s + 1.30·57-s + 0.660·59-s − 0.944·61-s + 0.726·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9221637453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9221637453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 + 4.10T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 + 3.39T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 3.25T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.567893317173075546641462471772, −7.73193017935738253610826282177, −6.53309197248514098599354219757, −6.31264785116444529283395868561, −5.63976296750575586509221207143, −4.77329998434688235527370987883, −4.15393535370905240442541746576, −2.86023475852019576797050357065, −1.86838455247738737990225607903, −0.57903530085571787698534637711,
0.57903530085571787698534637711, 1.86838455247738737990225607903, 2.86023475852019576797050357065, 4.15393535370905240442541746576, 4.77329998434688235527370987883, 5.63976296750575586509221207143, 6.31264785116444529283395868561, 6.53309197248514098599354219757, 7.73193017935738253610826282177, 8.567893317173075546641462471772
