| L(s) = 1 | − 3.34i·5-s + 7.19·13-s + 4.00i·17-s − 6.19·25-s + 10.6i·29-s + 11.3·37-s + 12.7i·41-s + 7·49-s + 12.7i·53-s + 5.39·61-s − 24.0i·65-s − 13.1·73-s + 13.3·85-s + 18.0i·89-s − 8·97-s + ⋯ |
| L(s) = 1 | − 1.49i·5-s + 1.99·13-s + 0.970i·17-s − 1.23·25-s + 1.98i·29-s + 1.87·37-s + 1.98i·41-s + 49-s + 1.74i·53-s + 0.690·61-s − 2.98i·65-s − 1.54·73-s + 1.45·85-s + 1.91i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.161708806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.161708806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.34iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.19T + 13T^{2} \) |
| 17 | \( 1 - 4.00iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 18.0iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.310985915596244589057678570691, −7.77847437072231081857529100485, −6.60723771593745207038136975069, −5.96135195100388107437331867126, −5.36851200526506299167173925973, −4.41540346631628875617819421401, −3.94266072376654101774795920924, −2.92563254349373294463243580490, −1.41017383725277621549410799926, −1.12998799145291547729561893251,
0.68500827244781171931718963901, 2.10303912291642183599488837109, 2.85241532655712777592586736071, 3.68273997889458812582183680557, 4.25891699897628032849144607366, 5.60344334913881434955640151509, 6.10280929596538925738143814013, 6.75419734373634392491124078184, 7.41696864404580677143978896141, 8.125346844852348021968820869983
