| L(s) = 1 | − 2-s − 2.86·3-s + 4-s − 2.90·5-s + 2.86·6-s + 0.924·7-s − 8-s + 5.18·9-s + 2.90·10-s − 1.96·11-s − 2.86·12-s + 4.75·13-s − 0.924·14-s + 8.31·15-s + 16-s − 3.12·17-s − 5.18·18-s − 2.40·19-s − 2.90·20-s − 2.64·21-s + 1.96·22-s − 5.24·23-s + 2.86·24-s + 3.45·25-s − 4.75·26-s − 6.24·27-s + 0.924·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.65·3-s + 0.5·4-s − 1.30·5-s + 1.16·6-s + 0.349·7-s − 0.353·8-s + 1.72·9-s + 0.919·10-s − 0.591·11-s − 0.825·12-s + 1.31·13-s − 0.247·14-s + 2.14·15-s + 0.250·16-s − 0.757·17-s − 1.22·18-s − 0.552·19-s − 0.650·20-s − 0.577·21-s + 0.418·22-s − 1.09·23-s + 0.583·24-s + 0.690·25-s − 0.931·26-s − 1.20·27-s + 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 - 0.924T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 5.39T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 - 2.11T + 41T^{2} \) |
| 43 | \( 1 - 0.246T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 0.976T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 2.07T + 79T^{2} \) |
| 83 | \( 1 + 8.77T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−8.100915227577844783584669330137, −7.36322619171868082264715821438, −6.59833585670464573011840359224, −5.99776456081372839538148579562, −5.20977442775081019692080563260, −4.26671566594552100015252340260, −3.71255445433271262223717353174, −2.13694789134998071271297598733, −0.881122066402227543314111782456, 0,
0.881122066402227543314111782456, 2.13694789134998071271297598733, 3.71255445433271262223717353174, 4.26671566594552100015252340260, 5.20977442775081019692080563260, 5.99776456081372839538148579562, 6.59833585670464573011840359224, 7.36322619171868082264715821438, 8.100915227577844783584669330137
