| L(s) = 1 | + 1.41i·5-s + 8·7-s + 11.3i·11-s − 8·13-s − 12.7i·17-s + 32·19-s + 33.9i·23-s + 23·25-s + 43.8i·29-s + 40·31-s + 11.3i·35-s − 26·37-s − 66.4i·41-s + 16·43-s − 11.3i·47-s + ⋯ |
| L(s) = 1 | + 0.282i·5-s + 1.14·7-s + 1.02i·11-s − 0.615·13-s − 0.748i·17-s + 1.68·19-s + 1.47i·23-s + 0.920·25-s + 1.51i·29-s + 1.29·31-s + 0.323i·35-s − 0.702·37-s − 1.62i·41-s + 0.372·43-s − 0.240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71756 + 0.545907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71756 + 0.545907i\) |
| \(L(2)\) |
|
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 | \( 1 - 1.41iT - 25T^{2} \) |
| 7 | \( 1 - 8T + 49T^{2} \) |
| 11 | \( 1 - 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 32T + 361T^{2} \) |
| 23 | \( 1 - 33.9iT - 529T^{2} \) |
| 29 | \( 1 - 43.8iT - 841T^{2} \) |
| 31 | \( 1 - 40T + 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 66.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 22.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54T + 3.72e3T^{2} \) |
| 67 | \( 1 + 80T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 104T + 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80T + 9.40e3T^{2} \) |
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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
−11.77573640306890491862472138972, −10.78239442851622589664447545251, −9.809451934059631004786224760246, −8.902707788714713351844101681885, −7.50691016726881283971362010970, −7.17710914941775217992666762092, −5.37943848611211427157673225897, −4.70485010934867225212261895938, −3.05674292188426112124169545645, −1.52132533619619718139131827775,
1.06233566860172633270219552727, 2.79127935295078644708489562469, 4.39837943542885904225736414052, 5.31927160319917222933418062191, 6.50633289204799949658122530871, 7.925303180359409807771799719426, 8.396358905066786162626619244616, 9.613536871876935525288933808240, 10.66683558921038300058287256799, 11.52911778306570364466573447617
