| L(s) = 1 | − 2-s − 0.841·3-s + 4-s − 1.88·5-s + 0.841·6-s − 4.21·7-s − 8-s − 2.29·9-s + 1.88·10-s − 4.82·11-s − 0.841·12-s + 5.72·13-s + 4.21·14-s + 1.58·15-s + 16-s − 1.08·17-s + 2.29·18-s + 3.42·19-s − 1.88·20-s + 3.54·21-s + 4.82·22-s + 3.34·23-s + 0.841·24-s − 1.45·25-s − 5.72·26-s + 4.45·27-s − 4.21·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.485·3-s + 0.5·4-s − 0.841·5-s + 0.343·6-s − 1.59·7-s − 0.353·8-s − 0.764·9-s + 0.595·10-s − 1.45·11-s − 0.242·12-s + 1.58·13-s + 1.12·14-s + 0.408·15-s + 0.250·16-s − 0.264·17-s + 0.540·18-s + 0.785·19-s − 0.420·20-s + 0.773·21-s + 1.02·22-s + 0.697·23-s + 0.171·24-s − 0.291·25-s − 1.12·26-s + 0.856·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 0.841T + 3T^{2} \) |
| 5 | \( 1 + 1.88T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 + 0.675T + 53T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−8.058652510861011389364783112487, −7.48650155694245704601201898852, −6.55786055722555098506611634100, −6.03011484583660537322081839087, −5.34721827827565466762899019792, −4.11068067142874943928406677345, −3.12706766878895789577745561626, −2.75247599943544875636777458002, −0.890361686429159630730872481344, 0,
0.890361686429159630730872481344, 2.75247599943544875636777458002, 3.12706766878895789577745561626, 4.11068067142874943928406677345, 5.34721827827565466762899019792, 6.03011484583660537322081839087, 6.55786055722555098506611634100, 7.48650155694245704601201898852, 8.058652510861011389364783112487
