| L(s) = 1 | + 8.57·3-s − 22.1·7-s + 46.4·9-s − 27.1·11-s + 70.3·13-s + 73.3·17-s + 110.·19-s − 189.·21-s − 107.·23-s + 167.·27-s + 68.6·29-s + 137.·31-s − 232.·33-s − 60.3·37-s + 602.·39-s + 95.1·41-s + 501.·43-s − 439.·47-s + 147.·49-s + 628.·51-s + 286.·53-s + 943.·57-s + 547.·59-s + 511.·61-s − 1.02e3·63-s + 301.·67-s − 924.·69-s + ⋯ |
| L(s) = 1 | + 1.64·3-s − 1.19·7-s + 1.72·9-s − 0.743·11-s + 1.50·13-s + 1.04·17-s + 1.32·19-s − 1.97·21-s − 0.977·23-s + 1.19·27-s + 0.439·29-s + 0.794·31-s − 1.22·33-s − 0.267·37-s + 2.47·39-s + 0.362·41-s + 1.77·43-s − 1.36·47-s + 0.429·49-s + 1.72·51-s + 0.743·53-s + 2.19·57-s + 1.20·59-s + 1.07·61-s − 2.05·63-s + 0.549·67-s − 1.61·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.665327408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.665327408\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8.57T + 27T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 11 | \( 1 + 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 68.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 137.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 60.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 286.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 547.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 301.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 82.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 704.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 743.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−9.834142670842808437334883738521, −8.996835544803422420229633572196, −8.184622468164126650608633646950, −7.60136117988080002921681314886, −6.51323909273190285989106661552, −5.51314532281432818786067571164, −3.90405185047841003700219860274, −3.32133230128389969787543840573, −2.51676929386042629931610906622, −1.04093935620705369036404150686,
1.04093935620705369036404150686, 2.51676929386042629931610906622, 3.32133230128389969787543840573, 3.90405185047841003700219860274, 5.51314532281432818786067571164, 6.51323909273190285989106661552, 7.60136117988080002921681314886, 8.184622468164126650608633646950, 8.996835544803422420229633572196, 9.834142670842808437334883738521
