L(s) = 1 | + 3i·3-s − 8i·7-s − 9·9-s − 36·11-s + 10i·13-s + 18i·17-s − 100·19-s + 24·21-s + 72i·23-s − 27i·27-s + 234·29-s + 16·31-s − 108i·33-s − 226i·37-s − 30·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.431i·7-s − 0.333·9-s − 0.986·11-s + 0.213i·13-s + 0.256i·17-s − 1.20·19-s + 0.249·21-s + 0.652i·23-s − 0.192i·27-s + 1.49·29-s + 0.0926·31-s − 0.569i·33-s − 1.00i·37-s − 0.123·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.418380403\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418380403\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 8iT - 343T^{2} \) |
| 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 234T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16T + 2.97e4T^{2} \) |
| 37 | \( 1 + 226iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 90T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 432iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 414iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 684T + 2.05e5T^{2} \) |
| 61 | \( 1 - 422T + 2.26e5T^{2} \) |
| 67 | \( 1 + 332iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 360T + 3.57e5T^{2} \) |
| 73 | \( 1 + 26iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 512T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 630T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3iT - 9.12e5T^{2} \) |
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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
−9.381881892869933960182162013941, −8.475333081674972906126037336195, −7.81782100322514914359738800832, −6.79196205248417130059729620231, −5.90177555109662382626326301929, −4.91573899118381531620797614928, −4.17885844787549614717750037822, −3.14021343648466992500253467655, −2.03677823919200040711537609870, −0.44215439147944638013897891276,
0.801574878833457261017641122618, 2.24664503001311452911475015496, 2.90007852801176338547779338095, 4.34449442111824018043152217349, 5.25821360304560886486595253284, 6.17198739622586690176000977699, 6.89505289257423392843508682470, 7.945042911459347257143714997860, 8.433470868779123943136003481536, 9.315585791559626645718932387939