| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2121697071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2121697071\) |
| \(L(\frac{5}{2})\) |
|
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
−9.451135384613268547906493220800, −8.105361315821050648086028905226, −7.11482787950398250615270177754, −6.41075824558867491020914530073, −6.00141043951628554723590918363, −4.98184357797265769649973277878, −4.24484651038009969344053185853, −3.08433039368316614348195834620, −1.63868256192472045443007041981, −0.23992551502397239403102950760,
0.23992551502397239403102950760, 1.63868256192472045443007041981, 3.08433039368316614348195834620, 4.24484651038009969344053185853, 4.98184357797265769649973277878, 6.00141043951628554723590918363, 6.41075824558867491020914530073, 7.11482787950398250615270177754, 8.105361315821050648086028905226, 9.451135384613268547906493220800
