| L(s) = 1 | − 2.78·2-s − 3-s + 5.73·4-s + 2.04·5-s + 2.78·6-s − 0.793·7-s − 10.3·8-s + 9-s − 5.68·10-s + 1.06·11-s − 5.73·12-s + 3.98·13-s + 2.20·14-s − 2.04·15-s + 17.4·16-s − 17-s − 2.78·18-s − 5.59·19-s + 11.7·20-s + 0.793·21-s − 2.95·22-s + 3.28·23-s + 10.3·24-s − 0.821·25-s − 11.0·26-s − 27-s − 4.54·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.577·3-s + 2.86·4-s + 0.914·5-s + 1.13·6-s − 0.299·7-s − 3.67·8-s + 0.333·9-s − 1.79·10-s + 0.320·11-s − 1.65·12-s + 1.10·13-s + 0.589·14-s − 0.527·15-s + 4.35·16-s − 0.242·17-s − 0.655·18-s − 1.28·19-s + 2.62·20-s + 0.173·21-s − 0.629·22-s + 0.683·23-s + 2.12·24-s − 0.164·25-s − 2.17·26-s − 0.192·27-s − 0.859·28-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 | \( 1 + 2.78T + 2T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 + 0.793T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 - 3.98T + 13T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 1.66T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.684T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 + 2.01T + 61T^{2} \) |
| 67 | \( 1 + 6.63T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−7.59432705899267966133386159468, −6.60333597108920502874407304770, −6.48908674391351952039896216248, −5.94440682122075342626579343576, −4.90928492422340332902305727097, −3.58183529213442800522938556459, −2.66766661712780976162528244276, −1.76480668533037768579834949589, −1.16325436394722654962578535137, 0,
1.16325436394722654962578535137, 1.76480668533037768579834949589, 2.66766661712780976162528244276, 3.58183529213442800522938556459, 4.90928492422340332902305727097, 5.94440682122075342626579343576, 6.48908674391351952039896216248, 6.60333597108920502874407304770, 7.59432705899267966133386159468
