| L(s) = 1 | + 1.57·2-s − 5.53·4-s − 5·5-s + 34.2·7-s − 21.2·8-s − 7.85·10-s − 26.8·11-s − 19.4·13-s + 53.7·14-s + 10.9·16-s + 29.1·17-s + 47.1·19-s + 27.6·20-s − 42.2·22-s + 113.·23-s + 25·25-s − 30.6·26-s − 189.·28-s + 81.2·29-s − 10.8·31-s + 187.·32-s + 45.7·34-s − 171.·35-s + 410.·37-s + 73.9·38-s + 106.·40-s − 443.·41-s + ⋯ |
| L(s) = 1 | + 0.555·2-s − 0.691·4-s − 0.447·5-s + 1.84·7-s − 0.939·8-s − 0.248·10-s − 0.737·11-s − 0.415·13-s + 1.02·14-s + 0.170·16-s + 0.415·17-s + 0.568·19-s + 0.309·20-s − 0.409·22-s + 1.02·23-s + 0.200·25-s − 0.230·26-s − 1.27·28-s + 0.520·29-s − 0.0627·31-s + 1.03·32-s + 0.230·34-s − 0.826·35-s + 1.82·37-s + 0.315·38-s + 0.420·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.194216716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194216716\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 2 | \( 1 - 1.57T + 8T^{2} \) |
| 7 | \( 1 - 34.2T + 343T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 81.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 10.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 443.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 609.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 15.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 61.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 216.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 65.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 522.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 801.T + 9.12e5T^{2} \) |
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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
−11.05107129385211003467575620361, −9.987023462173171780453155600818, −8.792799244282510171176000821966, −8.087637699293025828699597807943, −7.29971375068235128540608158173, −5.55567766833519892440067741919, −4.95335014449579350470096944370, −4.13546620660020685180015423952, −2.69859825181607269717792927407, −0.935079438675159345548818508351,
0.935079438675159345548818508351, 2.69859825181607269717792927407, 4.13546620660020685180015423952, 4.95335014449579350470096944370, 5.55567766833519892440067741919, 7.29971375068235128540608158173, 8.087637699293025828699597807943, 8.792799244282510171176000821966, 9.987023462173171780453155600818, 11.05107129385211003467575620361
