| L(s) = 1 | − 1.17·5-s − i·7-s − 2.90i·11-s + 1.05i·13-s + 6.48i·17-s + 3.11·19-s − 0.812·23-s − 3.61·25-s + 4.03·29-s − 0.108i·31-s + 1.17i·35-s − 9.97i·37-s + 9.44i·41-s − 10.3·43-s − 4.03·47-s + ⋯ |
| L(s) = 1 | − 0.526·5-s − 0.377i·7-s − 0.876i·11-s + 0.292i·13-s + 1.57i·17-s + 0.714·19-s − 0.169·23-s − 0.722·25-s + 0.749·29-s − 0.0195i·31-s + 0.199i·35-s − 1.63i·37-s + 1.47i·41-s − 1.57·43-s − 0.588·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0828 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057271011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057271011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.17T + 5T^{2} \) |
| 11 | \( 1 + 2.90iT - 11T^{2} \) |
| 13 | \( 1 - 1.05iT - 13T^{2} \) |
| 17 | \( 1 - 6.48iT - 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 + 0.812T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 + 0.108iT - 31T^{2} \) |
| 37 | \( 1 + 9.97iT - 37T^{2} \) |
| 41 | \( 1 - 9.44iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 2.97T + 53T^{2} \) |
| 59 | \( 1 + 0.868iT - 59T^{2} \) |
| 61 | \( 1 - 5.33iT - 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 0.844T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 + 9.48iT - 79T^{2} \) |
| 83 | \( 1 - 2.71iT - 83T^{2} \) |
| 89 | \( 1 + 2.22iT - 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−8.153920267951591608022466708081, −7.71376838788091425315326300518, −6.77934480003737724638483976199, −6.16012045553411408018639081787, −5.45017078167064500823556999050, −4.48942404358635033072750408581, −3.76979172344300025426617889544, −3.22963114629266880974431903349, −2.00359049217245490269909502083, −0.972840722720228416578872356050,
0.31274793646162925862883135415, 1.62171482472849614912161129274, 2.66986402244329778952415325506, 3.35305028477701887244192953364, 4.34448076326000255121507122358, 5.03514590187458441304802583756, 5.59401085299399394168617553827, 6.79414969418784190018753165118, 7.05584508813969281309123816400, 8.020930183977895595594495901126
