| L(s) = 1 | + 3.19·3-s − 2.92·5-s − 4.80·7-s + 7.22·9-s − 5.38·11-s + 3.67·13-s − 9.36·15-s − 1.72·17-s + 7.21·19-s − 15.3·21-s − 4.61·23-s + 3.56·25-s + 13.5·27-s + 8.38·29-s + 2.29·31-s − 17.2·33-s + 14.0·35-s − 1.44·37-s + 11.7·39-s + 2.85·41-s + 8.84·43-s − 21.1·45-s − 5.70·47-s + 16.0·49-s − 5.51·51-s + 10.4·53-s + 15.7·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 1.30·5-s − 1.81·7-s + 2.40·9-s − 1.62·11-s + 1.01·13-s − 2.41·15-s − 0.418·17-s + 1.65·19-s − 3.34·21-s − 0.961·23-s + 0.713·25-s + 2.60·27-s + 1.55·29-s + 0.412·31-s − 3.00·33-s + 2.37·35-s − 0.237·37-s + 1.88·39-s + 0.446·41-s + 1.34·43-s − 3.15·45-s − 0.831·47-s + 2.29·49-s − 0.772·51-s + 1.43·53-s + 2.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186925979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186925979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 2.80T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 + 2.35T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 3.06T + 97T^{2} \) |
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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
−8.328546449607389703759545904595, −7.902826213912923928725770145017, −7.22779559006671951181121853007, −6.58303289147154112235822379376, −5.41244875534596488952043615410, −4.11784464602442620373552700094, −3.70798322547317338947996446459, −2.94215844516425330410610797176, −2.55395186234072014398113225241, −0.75089331920601669324232082907,
0.75089331920601669324232082907, 2.55395186234072014398113225241, 2.94215844516425330410610797176, 3.70798322547317338947996446459, 4.11784464602442620373552700094, 5.41244875534596488952043615410, 6.58303289147154112235822379376, 7.22779559006671951181121853007, 7.902826213912923928725770145017, 8.328546449607389703759545904595
