| L(s) = 1 | + 2.45·2-s − 3-s + 4.02·4-s + 0.940·5-s − 2.45·6-s − 2.20·7-s + 4.97·8-s + 9-s + 2.30·10-s − 2.19·11-s − 4.02·12-s + 1.05·13-s − 5.41·14-s − 0.940·15-s + 4.16·16-s − 17-s + 2.45·18-s + 2.34·19-s + 3.78·20-s + 2.20·21-s − 5.38·22-s − 4.57·23-s − 4.97·24-s − 4.11·25-s + 2.59·26-s − 27-s − 8.87·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.01·4-s + 0.420·5-s − 1.00·6-s − 0.833·7-s + 1.76·8-s + 0.333·9-s + 0.730·10-s − 0.660·11-s − 1.16·12-s + 0.292·13-s − 1.44·14-s − 0.242·15-s + 1.04·16-s − 0.242·17-s + 0.578·18-s + 0.536·19-s + 0.846·20-s + 0.480·21-s − 1.14·22-s − 0.953·23-s − 1.01·24-s − 0.823·25-s + 0.508·26-s − 0.192·27-s − 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 - 0.940T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 0.413T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 - 0.440T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 + 7.41T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 + 0.0466T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.20055464654104630757759300878, −6.37327086389044918960067232225, −5.96729138475968620506380185304, −5.44564110930225799524168152785, −4.76590229949966091599022128240, −3.92332297033576902354624189403, −3.36041442513747194346174753637, −2.48793601725034665271303630822, −1.67482977704844313402489572025, 0,
1.67482977704844313402489572025, 2.48793601725034665271303630822, 3.36041442513747194346174753637, 3.92332297033576902354624189403, 4.76590229949966091599022128240, 5.44564110930225799524168152785, 5.96729138475968620506380185304, 6.37327086389044918960067232225, 7.20055464654104630757759300878
