| L(s) = 1 | + 0.713·3-s + 5-s − 0.376·7-s − 2.49·9-s + 2.82·13-s + 0.713·15-s + 4.83·17-s − 2.92·19-s − 0.268·21-s + 3.77·23-s + 25-s − 3.91·27-s − 8.44·29-s + 8.04·31-s − 0.376·35-s + 2.83·37-s + 2.01·39-s + 3.72·41-s + 6.48·43-s − 2.49·45-s + 2.58·47-s − 6.85·49-s + 3.45·51-s − 0.0487·53-s − 2.09·57-s − 1.64·59-s − 8.69·61-s + ⋯ |
| L(s) = 1 | + 0.411·3-s + 0.447·5-s − 0.142·7-s − 0.830·9-s + 0.784·13-s + 0.184·15-s + 1.17·17-s − 0.672·19-s − 0.0585·21-s + 0.786·23-s + 0.200·25-s − 0.753·27-s − 1.56·29-s + 1.44·31-s − 0.0635·35-s + 0.466·37-s + 0.323·39-s + 0.581·41-s + 0.988·43-s − 0.371·45-s + 0.377·47-s − 0.979·49-s + 0.483·51-s − 0.00669·53-s − 0.276·57-s − 0.214·59-s − 1.11·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.393461120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393461120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.713T + 3T^{2} \) |
| 7 | \( 1 + 0.376T + 7T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 - 3.77T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 - 8.04T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 0.0487T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 + 8.69T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 0.513T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.226947047629162120407115068762, −7.75861428259126171842585213434, −6.77316698034922705068731689493, −5.96922748866064565262647686867, −5.57189712646362881919700271116, −4.53479404553584763939496762084, −3.56385149276696390157566926774, −2.92640824561750650786425546662, −2.01063009561743829777229759639, −0.843675519415303466032959251363,
0.843675519415303466032959251363, 2.01063009561743829777229759639, 2.92640824561750650786425546662, 3.56385149276696390157566926774, 4.53479404553584763939496762084, 5.57189712646362881919700271116, 5.96922748866064565262647686867, 6.77316698034922705068731689493, 7.75861428259126171842585213434, 8.226947047629162120407115068762
