| L(s) = 1 | + (−1.22 + 1.22i)3-s + 2.44i·7-s − 2.99i·9-s + 4.89·11-s + 2·13-s − 6i·17-s − 4.89i·19-s + (−2.99 − 2.99i)21-s + 2.44·23-s + (3.67 + 3.67i)27-s + 9.79i·31-s + (−5.99 + 5.99i)33-s − 2·37-s + (−2.44 + 2.44i)39-s + 6i·41-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)3-s + 0.925i·7-s − 0.999i·9-s + 1.47·11-s + 0.554·13-s − 1.45i·17-s − 1.12i·19-s + (−0.654 − 0.654i)21-s + 0.510·23-s + (0.707 + 0.707i)27-s + 1.75i·31-s + (−1.04 + 1.04i)33-s − 0.328·37-s + (−0.392 + 0.392i)39-s + 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.428126919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428126919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7.34iT - 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4.89iT - 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.672045367067930824593610630285, −9.097902827134055229092963435242, −8.651863972942528703005729111691, −7.00776086012816722965923251691, −6.55757142543924919475344300470, −5.48198343971334385019764684463, −4.85438075173400318834763113047, −3.80049165169194238548802076759, −2.76628894480770565724675399282, −1.02265928121029032497586413324,
0.953227584013924735430808775520, 1.87429084522762548668632244192, 3.73689878620226175120083138195, 4.27796491866144545188912360587, 5.76849217771663929498867791674, 6.24768064334008538503096196507, 7.10569039448401730293426636740, 7.85904609877342988318968470675, 8.713472849883798957384161808416, 9.759286938498054559091843936905
