| L(s) = 1 | − 0.539·3-s − 2.17·5-s + 7-s − 2.70·9-s + 2.24·11-s + 13-s + 1.17·15-s − 5.80·17-s + 4.17·19-s − 0.539·21-s + 1.34·23-s − 0.290·25-s + 3.07·27-s − 2.41·29-s + 5.14·31-s − 1.21·33-s − 2.17·35-s + 8.58·37-s − 0.539·39-s − 5.41·41-s + 3.04·43-s + 5.87·45-s − 11.9·47-s + 49-s + 3.12·51-s + 9.20·53-s − 4.87·55-s + ⋯ |
| L(s) = 1 | − 0.311·3-s − 0.970·5-s + 0.377·7-s − 0.903·9-s + 0.677·11-s + 0.277·13-s + 0.302·15-s − 1.40·17-s + 0.956·19-s − 0.117·21-s + 0.279·23-s − 0.0581·25-s + 0.592·27-s − 0.449·29-s + 0.923·31-s − 0.211·33-s − 0.366·35-s + 1.41·37-s − 0.0863·39-s − 0.846·41-s + 0.465·43-s + 0.876·45-s − 1.73·47-s + 0.142·49-s + 0.437·51-s + 1.26·53-s − 0.657·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 1.34T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 - 8.58T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 3.04T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.20T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 - 2.49T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 - 0.219T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 8.06T + 97T^{2} \) |
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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
−7.929884081876718297589046969014, −6.96509743628995651732958537088, −6.41911416525764302970894383537, −5.58728296953493762022812535523, −4.77519119717132623909158186544, −4.11557853823958635811256600176, −3.32668875344304186358097649656, −2.41379908867590774887455789101, −1.14584537109285718151662143573, 0,
1.14584537109285718151662143573, 2.41379908867590774887455789101, 3.32668875344304186358097649656, 4.11557853823958635811256600176, 4.77519119717132623909158186544, 5.58728296953493762022812535523, 6.41911416525764302970894383537, 6.96509743628995651732958537088, 7.929884081876718297589046969014
