| L(s) = 1 | − 5.83i·5-s + 8.83i·7-s + 23.6·11-s − 54.6·13-s + 117. i·17-s + 109. i·19-s + 33.5·23-s + 90.9·25-s + 40.0i·29-s − 292. i·31-s + 51.5·35-s − 283.·37-s − 367. i·41-s − 323. i·43-s − 66.2·47-s + ⋯ |
| L(s) = 1 | − 0.522i·5-s + 0.476i·7-s + 0.647·11-s − 1.16·13-s + 1.67i·17-s + 1.32i·19-s + 0.304·23-s + 0.727·25-s + 0.256i·29-s − 1.69i·31-s + 0.249·35-s − 1.25·37-s − 1.39i·41-s − 1.14i·43-s − 0.205·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1914485954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1914485954\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.83iT - 125T^{2} \) |
| 7 | \( 1 - 8.83iT - 343T^{2} \) |
| 11 | \( 1 - 23.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 109. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 33.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 292. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 367. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 323. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 66.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 158. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 848.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 348.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 194. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 939.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 473.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 338.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 739. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 448.T + 9.12e5T^{2} \) |
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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
−9.182025252878453396712158409671, −8.679189155153670305187537919902, −7.84686092670259311897841377051, −7.00724223621292706266944861407, −5.99195826908811633702230757200, −5.41118314948191144624817538229, −4.34112052808061050946483911078, −3.60231304148671391747830311345, −2.26881522329052005286838142345, −1.40686949055077822998912849236,
0.04149202180676792522108580899, 1.21543364178056087309040932329, 2.67252474962996470487943509906, 3.21023684177395838741691068849, 4.68597688161096843177914464267, 4.96201084296621079293430541043, 6.41063292307893946454742018930, 7.08933923261451522869631278477, 7.42960905948003264238795559968, 8.732881953252267493939209737281
