| L(s) = 1 | − 1.11·2-s − 2.85·3-s − 0.757·4-s + 3.45·5-s + 3.18·6-s − 7-s + 3.07·8-s + 5.17·9-s − 3.85·10-s + 2.16·12-s + 2.05·13-s + 1.11·14-s − 9.88·15-s − 1.90·16-s − 1.93·17-s − 5.76·18-s − 1.62·19-s − 2.61·20-s + 2.85·21-s − 0.807·23-s − 8.78·24-s + 6.94·25-s − 2.29·26-s − 6.21·27-s + 0.757·28-s − 7.97·29-s + 11.0·30-s + ⋯ |
| L(s) = 1 | − 0.788·2-s − 1.65·3-s − 0.378·4-s + 1.54·5-s + 1.30·6-s − 0.377·7-s + 1.08·8-s + 1.72·9-s − 1.21·10-s + 0.625·12-s + 0.571·13-s + 0.297·14-s − 2.55·15-s − 0.477·16-s − 0.468·17-s − 1.35·18-s − 0.372·19-s − 0.585·20-s + 0.623·21-s − 0.168·23-s − 1.79·24-s + 1.38·25-s − 0.450·26-s − 1.19·27-s + 0.143·28-s − 1.48·29-s + 2.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6092613866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6092613866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + 0.807T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 0.788T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 7.56T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 + 3.18T + 71T^{2} \) |
| 73 | \( 1 + 1.22T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 4.43T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−10.22762843717305098949656353011, −9.496292920262044279212566798587, −8.875997782079879873038440174491, −7.52505839969330843603007683156, −6.48027400970915001598558460340, −5.90120331146382611749777341988, −5.16757058580961964353084090483, −4.12924256022665375678569056577, −2.01822243941629311599746089729, −0.792427191380879192946873505470,
0.792427191380879192946873505470, 2.01822243941629311599746089729, 4.12924256022665375678569056577, 5.16757058580961964353084090483, 5.90120331146382611749777341988, 6.48027400970915001598558460340, 7.52505839969330843603007683156, 8.875997782079879873038440174491, 9.496292920262044279212566798587, 10.22762843717305098949656353011
