| L(s) = 1 | + 2-s + 1.26·3-s + 4-s + 2.59·5-s + 1.26·6-s − 3.53·7-s + 8-s − 1.40·9-s + 2.59·10-s − 4.00·11-s + 1.26·12-s + 0.414·13-s − 3.53·14-s + 3.27·15-s + 16-s − 7.04·17-s − 1.40·18-s − 0.884·19-s + 2.59·20-s − 4.45·21-s − 4.00·22-s − 6.35·23-s + 1.26·24-s + 1.75·25-s + 0.414·26-s − 5.56·27-s − 3.53·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.728·3-s + 0.5·4-s + 1.16·5-s + 0.514·6-s − 1.33·7-s + 0.353·8-s − 0.469·9-s + 0.822·10-s − 1.20·11-s + 0.364·12-s + 0.115·13-s − 0.944·14-s + 0.846·15-s + 0.250·16-s − 1.70·17-s − 0.332·18-s − 0.202·19-s + 0.581·20-s − 0.973·21-s − 0.853·22-s − 1.32·23-s + 0.257·24-s + 0.351·25-s + 0.0813·26-s − 1.07·27-s − 0.668·28-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 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 + 0.884T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 + 0.466T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 0.538T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 5.43T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.103T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 9.99T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 9.03T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.189080846031618292721502681195, −7.16072247511335294001731421778, −6.35388854700935251645406344196, −5.97140095833857354217553659866, −5.17224426143107568242711397442, −4.16361932378792375047750329112, −3.24080288306066448477584859775, −2.49030669943480550292631389761, −2.08402528322694692549247991181, 0,
2.08402528322694692549247991181, 2.49030669943480550292631389761, 3.24080288306066448477584859775, 4.16361932378792375047750329112, 5.17224426143107568242711397442, 5.97140095833857354217553659866, 6.35388854700935251645406344196, 7.16072247511335294001731421778, 8.189080846031618292721502681195
