| L(s) = 1 | + 2-s + 1.99·3-s + 4-s − 3.74·5-s + 1.99·6-s − 0.855·7-s + 8-s + 0.990·9-s − 3.74·10-s + 0.952·11-s + 1.99·12-s − 1.62·13-s − 0.855·14-s − 7.48·15-s + 16-s + 2.22·17-s + 0.990·18-s − 2.08·19-s − 3.74·20-s − 1.70·21-s + 0.952·22-s + 3.03·23-s + 1.99·24-s + 9.04·25-s − 1.62·26-s − 4.01·27-s − 0.855·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s − 1.67·5-s + 0.815·6-s − 0.323·7-s + 0.353·8-s + 0.330·9-s − 1.18·10-s + 0.287·11-s + 0.576·12-s − 0.451·13-s − 0.228·14-s − 1.93·15-s + 0.250·16-s + 0.538·17-s + 0.233·18-s − 0.478·19-s − 0.837·20-s − 0.372·21-s + 0.203·22-s + 0.633·23-s + 0.407·24-s + 1.80·25-s − 0.319·26-s − 0.772·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 1.99T + 3T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 + 0.855T + 7T^{2} \) |
| 11 | \( 1 - 0.952T + 11T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 - 4.50T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 + 8.99T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 + 4.00T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 1.56T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.52953873876244617799755026929, −7.03091468808086529601836383748, −6.17204241633547927570164427393, −5.18689917824258497017752517827, −4.30009371234714776742970521561, −3.94603439481307036749104976662, −2.98740177046459422736977748562, −2.86354023399832253498848959044, −1.45168033088722635410272125733, 0,
1.45168033088722635410272125733, 2.86354023399832253498848959044, 2.98740177046459422736977748562, 3.94603439481307036749104976662, 4.30009371234714776742970521561, 5.18689917824258497017752517827, 6.17204241633547927570164427393, 7.03091468808086529601836383748, 7.52953873876244617799755026929
