| L(s) = 1 | + i·2-s − 4-s − 2i·7-s − i·8-s − 4·11-s + i·13-s + 2·14-s + 16-s − 4i·17-s + 2·19-s − 4i·22-s + 2i·23-s − 26-s + 2i·28-s + 8·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s − 1.20·11-s + 0.277i·13-s + 0.534·14-s + 0.250·16-s − 0.970i·17-s + 0.458·19-s − 0.852i·22-s + 0.417i·23-s − 0.196·26-s + 0.377i·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4562551327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4562551327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.81027865582617784190063839661, −7.16346118872494318304047449206, −6.61704181109130251357099700072, −5.73713270730013353725998647163, −4.89974354996072214345556864808, −4.55496172354194629201673658959, −3.38183559743528123590714628350, −2.69421407146585450526738880090, −1.28126572348128815560942206798, −0.12626232242913018570040288438,
1.23154781727422314673933464467, 2.37111631717647593122169012166, 2.85700208559530177837165812390, 3.78442418218677366242854209576, 4.76510364973316533840518335260, 5.32474704536153101986721066779, 6.05834599149989146360436572984, 6.87456431896450521290729338210, 8.051691390751707992494283534825, 8.207672857865422703948717488355
