| L(s) = 1 | − 1.65·2-s − 5.25·4-s + 5·5-s + 21.9·8-s − 8.28·10-s − 69.5·11-s + 68.4·13-s + 5.70·16-s − 104.·17-s + 71.8·19-s − 26.2·20-s + 115.·22-s + 101.·23-s + 25·25-s − 113.·26-s + 114.·29-s − 73.6·31-s − 185.·32-s + 172.·34-s − 200.·37-s − 119.·38-s + 109.·40-s − 417.·41-s + 311.·43-s + 365.·44-s − 167.·46-s − 149.·47-s + ⋯ |
| L(s) = 1 | − 0.585·2-s − 0.657·4-s + 0.447·5-s + 0.970·8-s − 0.261·10-s − 1.90·11-s + 1.45·13-s + 0.0890·16-s − 1.48·17-s + 0.868·19-s − 0.293·20-s + 1.11·22-s + 0.915·23-s + 0.200·25-s − 0.854·26-s + 0.734·29-s − 0.426·31-s − 1.02·32-s + 0.871·34-s − 0.892·37-s − 0.508·38-s + 0.433·40-s − 1.58·41-s + 1.10·43-s + 1.25·44-s − 0.536·46-s − 0.464·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.027173579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027173579\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 8T^{2} \) |
| 11 | \( 1 + 69.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 149.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 271.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 80.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 70.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 145.T + 9.12e5T^{2} \) |
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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
−8.640757468656464497898828162807, −8.212291563958438094038602850228, −7.30325371393633714733505691216, −6.43408156581506374296800911242, −5.30845347243766107706461890862, −4.95177163639709088807736015613, −3.77412055282495582050677978420, −2.74796657112291585281408525453, −1.61535225736761742405203726089, −0.51523972753236049132633325599,
0.51523972753236049132633325599, 1.61535225736761742405203726089, 2.74796657112291585281408525453, 3.77412055282495582050677978420, 4.95177163639709088807736015613, 5.30845347243766107706461890862, 6.43408156581506374296800911242, 7.30325371393633714733505691216, 8.212291563958438094038602850228, 8.640757468656464497898828162807
