| L(s) = 1 | − 0.546·2-s + 1.17·3-s − 1.70·4-s + 3.07·5-s − 0.640·6-s − 2.98·7-s + 2.02·8-s − 1.62·9-s − 1.67·10-s − 3.12·11-s − 1.99·12-s + 1.63·14-s + 3.60·15-s + 2.29·16-s + 17-s + 0.886·18-s + 0.0560·19-s − 5.23·20-s − 3.50·21-s + 1.70·22-s + 1.42·23-s + 2.37·24-s + 4.46·25-s − 5.42·27-s + 5.08·28-s − 1.95·29-s − 1.97·30-s + ⋯ |
| L(s) = 1 | − 0.386·2-s + 0.677·3-s − 0.850·4-s + 1.37·5-s − 0.261·6-s − 1.12·7-s + 0.714·8-s − 0.541·9-s − 0.531·10-s − 0.942·11-s − 0.576·12-s + 0.436·14-s + 0.931·15-s + 0.574·16-s + 0.242·17-s + 0.209·18-s + 0.0128·19-s − 1.17·20-s − 0.764·21-s + 0.363·22-s + 0.297·23-s + 0.484·24-s + 0.892·25-s − 1.04·27-s + 0.960·28-s − 0.362·29-s − 0.359·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.467891777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467891777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.546T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 19 | \( 1 - 0.0560T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 0.980T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−8.823410843768566513947703643286, −8.306823462468064607950144456005, −7.44830086722702698024594366641, −6.40039108077335443902172625966, −5.69783630010794515346851412374, −5.06613675800496365801098228568, −3.87535554210846228130972268015, −2.89397633993926498148741981086, −2.27414489785871939682553594616, −0.75461169795624258066828513726,
0.75461169795624258066828513726, 2.27414489785871939682553594616, 2.89397633993926498148741981086, 3.87535554210846228130972268015, 5.06613675800496365801098228568, 5.69783630010794515346851412374, 6.40039108077335443902172625966, 7.44830086722702698024594366641, 8.306823462468064607950144456005, 8.823410843768566513947703643286
