| L(s) = 1 | + 3.09i·3-s − 0.646i·5-s + (−2.44 − i)7-s − 6.58·9-s + (−2.79 − 1.79i)11-s − 3.09·13-s + 2·15-s + 3.74·17-s + 5.54·19-s + (3.09 − 7.58i)21-s − 4·23-s + 4.58·25-s − 11.0i·27-s − 7.58i·29-s + 1.15i·31-s + ⋯ |
| L(s) = 1 | + 1.78i·3-s − 0.288i·5-s + (−0.925 − 0.377i)7-s − 2.19·9-s + (−0.841 − 0.540i)11-s − 0.858·13-s + 0.516·15-s + 0.907·17-s + 1.27·19-s + (0.675 − 1.65i)21-s − 0.834·23-s + 0.916·25-s − 2.13i·27-s − 1.40i·29-s + 0.207i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5646734218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5646734218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
| 11 | \( 1 + (2.79 + 1.79i)T \) |
| good | 3 | \( 1 - 3.09iT - 3T^{2} \) |
| 5 | \( 1 + 0.646iT - 5T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.58iT - 29T^{2} \) |
| 31 | \( 1 - 1.15iT - 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 5.03T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 5.03iT - 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 + 3.09iT - 59T^{2} \) |
| 61 | \( 1 + 9.28T + 61T^{2} \) |
| 67 | \( 1 + 1.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−9.818863548378795982867827473626, −9.067015709466671720185747947008, −8.154917938988702495365321068842, −7.21434800936460653782970244501, −5.80149826066402213957365585456, −5.32712747280646337477438481908, −4.37529108923880265164623627220, −3.46415058075594768025444739144, −2.79113513482241949368269326517, −0.24332268518612180130058660661,
1.33522507788615557230166931425, 2.62899695660064372959094621524, 3.12975892632169872049164146607, 5.06229775418347563302877265659, 5.86630612758231716054704099654, 6.67544569849725591828448222045, 7.48043168846257916384044923576, 7.75660778050889558756898660157, 8.944921238773076961856405978468, 9.779305909570675418925256398091
