| L(s) = 1 | − 1.49i·5-s + 26.1i·7-s − 56.3·11-s + 41.3·13-s + 51.0i·17-s − 79.0i·19-s + 27.3·23-s + 122.·25-s + 134. i·29-s + 187. i·31-s + 39.1·35-s + 196.·37-s + 298. i·41-s − 465. i·43-s − 373.·47-s + ⋯ |
| L(s) = 1 | − 0.133i·5-s + 1.41i·7-s − 1.54·11-s + 0.881·13-s + 0.728i·17-s − 0.954i·19-s + 0.248·23-s + 0.982·25-s + 0.861i·29-s + 1.08i·31-s + 0.189·35-s + 0.873·37-s + 1.13i·41-s − 1.65i·43-s − 1.16·47-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
| \(L(2)\) |
\(\approx\) |
\(0.7084031654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7084031654\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.49iT - 125T^{2} \) |
| 7 | \( 1 - 26.1iT - 343T^{2} \) |
| 11 | \( 1 + 56.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 79.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 27.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 187. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 465. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 620. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 223.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 70.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.34e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 576.T + 9.12e5T^{2} \) |
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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
−9.009132162688497449787194359636, −8.723780345403998997285606529496, −7.955234992723438228110782400968, −6.92974188050854390227560641353, −6.00197191069077577198308613222, −5.32518815816803226851435324118, −4.63525132274263566351920054151, −3.16227420223062535388002468529, −2.58612718259247700296983376381, −1.37216324887801639381007361635,
0.16122126590392843008578906526, 1.13736796658922203685111261590, 2.51896664057513656690600197160, 3.49623157590581687057143839551, 4.36097889192103090363749269530, 5.24159528150964047093420353433, 6.20554743220510138293787738265, 7.06002982748023500909157291543, 7.86712360082485245907229229282, 8.246433525137805266748435160209
