| L(s) = 1 | + 5.71i·5-s + 5.91·7-s − 2.41i·11-s + 17.1·13-s − 16.1i·17-s + 4.35·19-s + 15.9i·23-s − 7.70·25-s + 31.2i·29-s − 13.5·31-s + 33.8i·35-s + 45.7·37-s + 30.3i·41-s − 21.0·43-s + 43.6i·47-s + ⋯ |
| L(s) = 1 | + 1.14i·5-s + 0.845·7-s − 0.219i·11-s + 1.31·13-s − 0.949i·17-s + 0.229·19-s + 0.694i·23-s − 0.308·25-s + 1.07i·29-s − 0.435·31-s + 0.966i·35-s + 1.23·37-s + 0.740i·41-s − 0.490·43-s + 0.928i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.114531190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114531190\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
| good | 5 | \( 1 - 5.71iT - 25T^{2} \) |
| 7 | \( 1 - 5.91T + 49T^{2} \) |
| 11 | \( 1 + 2.41iT - 121T^{2} \) |
| 13 | \( 1 - 17.1T + 169T^{2} \) |
| 17 | \( 1 + 16.1iT - 289T^{2} \) |
| 23 | \( 1 - 15.9iT - 529T^{2} \) |
| 29 | \( 1 - 31.2iT - 841T^{2} \) |
| 31 | \( 1 + 13.5T + 961T^{2} \) |
| 37 | \( 1 - 45.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 43.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 43.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 86.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 112.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 13.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 71.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 20.1T + 9.40e3T^{2} \) |
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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
−10.61936774129922558347281107491, −9.594235258810687322676913067542, −8.628484387571211220465062075488, −7.73008088990332466662290189386, −6.92403128612840088502802126526, −6.00939000099422869189436116233, −4.98412611916468983907298577984, −3.70098039049954510403106641379, −2.77110265359096898211499835035, −1.30707181656737745187155756851,
0.900649245790446282441797890995, 1.97963095329113809115040010902, 3.77584988945666706044352167897, 4.61209355158206716636488340663, 5.54240573049868575560372433236, 6.46394468160169625496774831626, 7.85313468420215433395069990569, 8.415549499166469618486688766150, 9.075869076649425309116298985622, 10.16409357007019825350960261878
