L(s) = 1 | + (1 + i)2-s + 2i·4-s − 2·7-s + (−2 + 2i)8-s + 4i·11-s + (−2 − 2i)14-s − 4·16-s − 6·17-s − 4i·19-s + (−4 + 4i)22-s − 4·23-s − 4i·28-s − 6i·29-s + 10·31-s + (−4 − 4i)32-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s − 0.755·7-s + (−0.707 + 0.707i)8-s + 1.20i·11-s + (−0.534 − 0.534i)14-s − 16-s − 1.45·17-s − 0.917i·19-s + (−0.852 + 0.852i)22-s − 0.834·23-s − 0.755i·28-s − 1.11i·29-s + 1.79·31-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6679904337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6679904337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 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
−9.671632871743074861267639454371, −8.840673059193597144670569822022, −8.085107724007899737535770948726, −7.09353897325785887548341595208, −6.63203457664901773289278052594, −5.90587953654790420190448298493, −4.62510904831370408658515021086, −4.36661479518166264265115554616, −3.05537090306069614944728035551, −2.19483216598064543239036093092,
0.17848268403632496330964072075, 1.69654807277499282291602406887, 2.90187725485257266717297813458, 3.58007478127175611038371481458, 4.50311049206574359300872126957, 5.49135307478256597520897035521, 6.33035114290747587140794483536, 6.75634869648055366208379328080, 8.249431749506320013405460716063, 8.831032880567445837712472912468