| L(s) = 1 | + (−1.10 + 1.91i)5-s + (0.906 − 2.48i)7-s + (−2.93 + 1.69i)11-s + (−1.56 − 0.901i)13-s + 5.96·17-s + 1.64i·19-s + (2.05 + 1.18i)23-s + (0.0556 + 0.0963i)25-s + (−2.44 + 1.41i)29-s + (−9.28 − 5.36i)31-s + (3.75 + 4.48i)35-s + 1.69·37-s + (0.455 − 0.788i)41-s + (1.96 + 3.39i)43-s + (0.123 + 0.213i)47-s + ⋯ |
| L(s) = 1 | + (−0.494 + 0.856i)5-s + (0.342 − 0.939i)7-s + (−0.885 + 0.511i)11-s + (−0.432 − 0.249i)13-s + 1.44·17-s + 0.377i·19-s + (0.428 + 0.247i)23-s + (0.0111 + 0.0192i)25-s + (−0.453 + 0.262i)29-s + (−1.66 − 0.962i)31-s + (0.635 + 0.757i)35-s + 0.279·37-s + (0.0710 − 0.123i)41-s + (0.299 + 0.517i)43-s + (0.0179 + 0.0310i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7808839220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7808839220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.906 + 2.48i)T \) |
| good | 5 | \( 1 + (1.10 - 1.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 + 0.901i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 - 1.64iT - 19T^{2} \) |
| 23 | \( 1 + (-2.05 - 1.18i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (9.28 + 5.36i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + (-0.455 + 0.788i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.96 - 3.39i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.123 - 0.213i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 7.87iT - 53T^{2} \) |
| 59 | \( 1 + (5.39 - 9.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 0.709i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 0.426iT - 73T^{2} \) |
| 79 | \( 1 + (2.49 + 4.31i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.28 + 7.42i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + (6.30 - 3.63i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.029326758476054373836743665605, −7.81065472112982342319252124696, −7.52662447705424758288784739866, −7.14033378587727041511532309171, −5.88150617685096209799928124761, −5.20346805386320937106143053122, −4.19412766318331350070060920475, −3.44627935786730278255520274841, −2.61322688616114764761634905024, −1.29517789398121792877648056714,
0.25314898104964844901998818479, 1.61739796210970527265636172204, 2.74635651403898931623134387260, 3.61502148572684288990687445542, 4.81808650674380834878127140272, 5.24531977417530023959950536749, 5.92393641634658486439448711947, 7.10735294789159457884506503983, 7.88636061371696600267461321339, 8.391812506443172921972651411178
