| L(s) = 1 | − 2-s + 0.388·3-s + 4-s − 3.74·5-s − 0.388·6-s − 4.20·7-s − 8-s − 2.84·9-s + 3.74·10-s + 0.0915·11-s + 0.388·12-s + 3.00·13-s + 4.20·14-s − 1.45·15-s + 16-s + 5.06·17-s + 2.84·18-s + 4.20·19-s − 3.74·20-s − 1.63·21-s − 0.0915·22-s + 3.33·23-s − 0.388·24-s + 9.01·25-s − 3.00·26-s − 2.27·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.224·3-s + 0.5·4-s − 1.67·5-s − 0.158·6-s − 1.58·7-s − 0.353·8-s − 0.949·9-s + 1.18·10-s + 0.0276·11-s + 0.112·12-s + 0.834·13-s + 1.12·14-s − 0.375·15-s + 0.250·16-s + 1.22·17-s + 0.671·18-s + 0.963·19-s − 0.837·20-s − 0.356·21-s − 0.0195·22-s + 0.695·23-s − 0.0793·24-s + 1.80·25-s − 0.589·26-s − 0.437·27-s − 0.794·28-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 | \( 1 - 0.388T + 3T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 - 0.0915T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 + 0.812T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.02T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 - 7.36T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 - 0.844T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.185892810803055103073676286677, −7.42622404370939667927103429825, −6.89960350797601492023522853277, −6.00921597000620937094179976354, −5.22894612705924746093776152160, −3.66500913775029850236588967927, −3.50606434853629879384413520555, −2.76937617364416000497042940170, −0.971902858685020803984390040302, 0,
0.971902858685020803984390040302, 2.76937617364416000497042940170, 3.50606434853629879384413520555, 3.66500913775029850236588967927, 5.22894612705924746093776152160, 6.00921597000620937094179976354, 6.89960350797601492023522853277, 7.42622404370939667927103429825, 8.185892810803055103073676286677
