| L(s) = 1 | − 2-s + 3-s + 4-s − 2.69·5-s − 6-s − 3.87·7-s − 8-s + 9-s + 2.69·10-s − 6.10·11-s + 12-s + 5.79·13-s + 3.87·14-s − 2.69·15-s + 16-s + 8.21·17-s − 18-s + 4.11·19-s − 2.69·20-s − 3.87·21-s + 6.10·22-s − 23-s − 24-s + 2.25·25-s − 5.79·26-s + 27-s − 3.87·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.20·5-s − 0.408·6-s − 1.46·7-s − 0.353·8-s + 0.333·9-s + 0.851·10-s − 1.84·11-s + 0.288·12-s + 1.60·13-s + 1.03·14-s − 0.695·15-s + 0.250·16-s + 1.99·17-s − 0.235·18-s + 0.944·19-s − 0.602·20-s − 0.846·21-s + 1.30·22-s − 0.208·23-s − 0.204·24-s + 0.451·25-s − 1.13·26-s + 0.192·27-s − 0.733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 - 8.21T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 6.65T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 7.30T + 97T^{2} \) |
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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
−8.179652530733629203584229613045, −7.54207439310351388837298797230, −6.95678425196377931917284626332, −5.92891937040524937261766939313, −5.25009901827942567626099290320, −3.73029235010573170638258859963, −3.38106894961160193091346018471, −2.73959649428372584792550574622, −1.14387340517247776199677169083, 0,
1.14387340517247776199677169083, 2.73959649428372584792550574622, 3.38106894961160193091346018471, 3.73029235010573170638258859963, 5.25009901827942567626099290320, 5.92891937040524937261766939313, 6.95678425196377931917284626332, 7.54207439310351388837298797230, 8.179652530733629203584229613045
