| L(s) = 1 | + 2.44i·3-s + 2.44·5-s + (2.44 − i)7-s − 2.99·9-s − 2·11-s − 2.44·13-s + 5.99i·15-s + 4.89i·17-s − 2.44i·19-s + (2.44 + 5.99i)21-s − 4i·23-s + 0.999·25-s − 4i·29-s − 4.89·31-s − 4.89i·33-s + ⋯ |
| L(s) = 1 | + 1.41i·3-s + 1.09·5-s + (0.925 − 0.377i)7-s − 0.999·9-s − 0.603·11-s − 0.679·13-s + 1.54i·15-s + 1.18i·17-s − 0.561i·19-s + (0.534 + 1.30i)21-s − 0.834i·23-s + 0.199·25-s − 0.742i·29-s − 0.879·31-s − 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21415 + 0.806797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21415 + 0.806797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.44 + i)T \) |
| good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 2.44iT - 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−12.44664358031908622717355607955, −10.92938390603178442395831411551, −10.55369660702786565206039681449, −9.664774405643186667400097549353, −8.805035719116370052326094200536, −7.55322411434236714029125661760, −5.91302053280658659872768452727, −5.00121435561395097778386467622, −4.05367319959355114966001506014, −2.26311276887018369188059368302,
1.57422341470797543883992671738, 2.58063938810333539765476774955, 5.06910515916291858241447876815, 5.88686964602350435292808038585, 7.16047148905987817876932267415, 7.84210809073346611461100875031, 9.043738392548493290056368442174, 10.10124702483204552487370157243, 11.37250155785059778882305688161, 12.20313345108792538167371349579
