| L(s) = 1 | + 3.14i·5-s − 3.44i·7-s − 4.56·11-s + 6.89·13-s + 3.46i·17-s + 4.89i·19-s − 2.82·23-s − 4.89·25-s − 2.19i·29-s + 2.55i·31-s + 10.8·35-s + 4.89·37-s + 4.09i·41-s + 2.89i·43-s + 2.19·47-s + ⋯ |
| L(s) = 1 | + 1.40i·5-s − 1.30i·7-s − 1.37·11-s + 1.91·13-s + 0.840i·17-s + 1.12i·19-s − 0.589·23-s − 0.979·25-s − 0.407i·29-s + 0.458i·31-s + 1.83·35-s + 0.805·37-s + 0.640i·41-s + 0.442i·43-s + 0.319·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419966063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419966063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.14iT - 5T^{2} \) |
| 7 | \( 1 + 3.44iT - 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 2.19iT - 29T^{2} \) |
| 31 | \( 1 - 2.55iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 4.09iT - 41T^{2} \) |
| 43 | \( 1 - 2.89iT - 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.89T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.02iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.900429735874353422952118743540, −8.449010709676954138896107970407, −7.87369788460388826029371485298, −7.20702994597106565309576660520, −6.24742469082687508108479281410, −5.81133107565770193077081844229, −4.23326907024666438766592742112, −3.64124188422156186342450232739, −2.73834587221283664102012434564, −1.33327382355754718703690081059,
0.56770975674730254520479892569, 1.97001597124107924997606638913, 2.99964976748049310157864451397, 4.26608400051538381394171792359, 5.28931870455206319304490502813, 5.53178914149489442237267755547, 6.58361082167563346850313676004, 7.88629393262715403277456088973, 8.459099993455520483534416973837, 8.989028221279246306259444890113
