| L(s) = 1 | − 2-s + 2.98·3-s + 4-s + 2.40·5-s − 2.98·6-s − 2.85·7-s − 8-s + 5.88·9-s − 2.40·10-s + 1.91·11-s + 2.98·12-s − 0.618·13-s + 2.85·14-s + 7.16·15-s + 16-s + 4.77·17-s − 5.88·18-s − 19-s + 2.40·20-s − 8.50·21-s − 1.91·22-s + 0.942·23-s − 2.98·24-s + 0.768·25-s + 0.618·26-s + 8.60·27-s − 2.85·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.72·3-s + 0.5·4-s + 1.07·5-s − 1.21·6-s − 1.07·7-s − 0.353·8-s + 1.96·9-s − 0.759·10-s + 0.576·11-s + 0.860·12-s − 0.171·13-s + 0.762·14-s + 1.84·15-s + 0.250·16-s + 1.15·17-s − 1.38·18-s − 0.229·19-s + 0.537·20-s − 1.85·21-s − 0.407·22-s + 0.196·23-s − 0.608·24-s + 0.153·25-s + 0.121·26-s + 1.65·27-s − 0.539·28-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
| \(L(1)\) |
\(\approx\) |
\(3.584988359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584988359\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 23 | \( 1 - 0.942T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 8.35T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 - 0.0157T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 4.77T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.227396340401370700515914379130, −7.05892494744331189782815288841, −6.81529423835422735019863005601, −5.99681927996481733025980367442, −5.08112726081982227405819997186, −3.84808935341705940921932412532, −3.27853253994023241629757832955, −2.59507830072847613138011877569, −1.91344212985781408942740415796, −0.995545351614309447777922952549,
0.995545351614309447777922952549, 1.91344212985781408942740415796, 2.59507830072847613138011877569, 3.27853253994023241629757832955, 3.84808935341705940921932412532, 5.08112726081982227405819997186, 5.99681927996481733025980367442, 6.81529423835422735019863005601, 7.05892494744331189782815288841, 8.227396340401370700515914379130
