| L(s) = 1 | + 2-s + 0.0462·3-s + 4-s + 0.819·5-s + 0.0462·6-s − 4.39·7-s + 8-s − 2.99·9-s + 0.819·10-s − 3.12·11-s + 0.0462·12-s − 0.685·13-s − 4.39·14-s + 0.0379·15-s + 16-s − 6.67·17-s − 2.99·18-s + 19-s + 0.819·20-s − 0.203·21-s − 3.12·22-s + 7.36·23-s + 0.0462·24-s − 4.32·25-s − 0.685·26-s − 0.277·27-s − 4.39·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0267·3-s + 0.5·4-s + 0.366·5-s + 0.0188·6-s − 1.66·7-s + 0.353·8-s − 0.999·9-s + 0.259·10-s − 0.942·11-s + 0.0133·12-s − 0.190·13-s − 1.17·14-s + 0.00979·15-s + 0.250·16-s − 1.61·17-s − 0.706·18-s + 0.229·19-s + 0.183·20-s − 0.0443·21-s − 0.666·22-s + 1.53·23-s + 0.00944·24-s − 0.865·25-s − 0.134·26-s − 0.0534·27-s − 0.830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650365930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650365930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0462T + 3T^{2} \) |
| 5 | \( 1 - 0.819T + 5T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 0.685T + 13T^{2} \) |
| 17 | \( 1 + 6.67T + 17T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 - 0.636T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−7.51791854550410711141414806310, −7.06467905931495942715928667074, −6.21374170146303880704228376043, −5.87024285391575795337297066244, −5.12669732995020449710476566981, −4.28585033999228481664605365039, −3.36922153621721927009180886655, −2.72143871765472289278050738991, −2.28009943933939567269174535667, −0.52432264476808064825601107762,
0.52432264476808064825601107762, 2.28009943933939567269174535667, 2.72143871765472289278050738991, 3.36922153621721927009180886655, 4.28585033999228481664605365039, 5.12669732995020449710476566981, 5.87024285391575795337297066244, 6.21374170146303880704228376043, 7.06467905931495942715928667074, 7.51791854550410711141414806310
