| L(s) = 1 | − 2.62·2-s − 3-s + 4.90·4-s + 1.61·5-s + 2.62·6-s − 0.284·7-s − 7.62·8-s + 9-s − 4.22·10-s − 2.96·11-s − 4.90·12-s − 0.885·13-s + 0.747·14-s − 1.61·15-s + 10.2·16-s + 17-s − 2.62·18-s + 3.78·19-s + 7.89·20-s + 0.284·21-s + 7.78·22-s + 7.24·23-s + 7.62·24-s − 2.40·25-s + 2.32·26-s − 27-s − 1.39·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.45·4-s + 0.720·5-s + 1.07·6-s − 0.107·7-s − 2.69·8-s + 0.333·9-s − 1.33·10-s − 0.893·11-s − 1.41·12-s − 0.245·13-s + 0.199·14-s − 0.415·15-s + 2.55·16-s + 0.242·17-s − 0.619·18-s + 0.867·19-s + 1.76·20-s + 0.0620·21-s + 1.65·22-s + 1.51·23-s + 1.55·24-s − 0.481·25-s + 0.456·26-s − 0.192·27-s − 0.263·28-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 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 0.284T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 0.885T + 13T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 0.699T + 29T^{2} \) |
| 31 | \( 1 + 9.02T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 - 0.732T + 41T^{2} \) |
| 43 | \( 1 + 7.42T + 43T^{2} \) |
| 47 | \( 1 - 0.637T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 2.66T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−7.52914027047332697885066082706, −7.02050036060787784674972082307, −6.41054821851361324786132874715, −5.41750724709479423381552671638, −5.22451220778737164328060945535, −3.57889155421177756107948251758, −2.66531806888297874539122324887, −1.89649984566834242730330736784, −1.03970715986089192986887029939, 0,
1.03970715986089192986887029939, 1.89649984566834242730330736784, 2.66531806888297874539122324887, 3.57889155421177756107948251758, 5.22451220778737164328060945535, 5.41750724709479423381552671638, 6.41054821851361324786132874715, 7.02050036060787784674972082307, 7.52914027047332697885066082706
