| L(s) = 1 | + 1.73i·5-s + (0.5 + 2.59i)7-s − 5.19i·11-s − 3.46i·13-s + 3.46i·17-s − 8·19-s + 6.92i·23-s + 2.00·25-s − 6·29-s + 5·31-s + (−4.5 + 0.866i)35-s + 4·37-s + 6.92i·41-s + 3.46i·43-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + (0.188 + 0.981i)7-s − 1.56i·11-s − 0.960i·13-s + 0.840i·17-s − 1.83·19-s + 1.44i·23-s + 0.400·25-s − 1.11·29-s + 0.898·31-s + (−0.760 + 0.146i)35-s + 0.657·37-s + 1.08i·41-s + 0.528i·43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7279197538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7279197538\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 2.59i)T \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.903266117358267562791217637225, −8.263261845729627526623236309300, −7.80719320203608736589171944391, −6.55698003161024981615174731036, −6.01327416704652530955086595439, −5.51530470794969172823236751208, −4.31867201960808820661074573095, −3.23941534116565580275448290458, −2.76508530729559748218254686301, −1.51640728743610610827217440154,
0.21782973325106660387588493676, 1.59928495155205625869217911580, 2.40549431616391595933096612878, 3.95050717694923147376805777503, 4.59042925351812226200300028385, 4.86876758626749927299003927798, 6.35346546022633633832504714052, 6.87456772062442379134848680355, 7.59129733737160161103192656522, 8.438209848893797957704546536977
