| L(s) = 1 | + (−1.00 + 5.09i)3-s + (13.2 − 4.82i)5-s + (3.17 + 17.9i)7-s + (−24.9 − 10.2i)9-s + (33.5 + 12.2i)11-s + (−6.15 − 5.16i)13-s + (11.2 + 72.3i)15-s + (−58.3 + 101. i)17-s + (39.9 + 69.1i)19-s + (−94.9 − 1.97i)21-s + (−15.4 + 87.8i)23-s + (56.5 − 47.4i)25-s + (77.5 − 116. i)27-s + (85.5 − 71.7i)29-s + (47.1 − 267. i)31-s + ⋯ |
| L(s) = 1 | + (−0.194 + 0.980i)3-s + (1.18 − 0.431i)5-s + (0.171 + 0.971i)7-s + (−0.924 − 0.380i)9-s + (0.919 + 0.334i)11-s + (−0.131 − 0.110i)13-s + (0.193 + 1.24i)15-s + (−0.832 + 1.44i)17-s + (0.481 + 0.834i)19-s + (−0.986 − 0.0205i)21-s + (−0.140 + 0.796i)23-s + (0.452 − 0.379i)25-s + (0.553 − 0.833i)27-s + (0.547 − 0.459i)29-s + (0.272 − 1.54i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.33671 + 1.11681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33671 + 1.11681i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.00 - 5.09i)T \) |
| good | 5 | \( 1 + (-13.2 + 4.82i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-3.17 - 17.9i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-33.5 - 12.2i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (6.15 + 5.16i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (58.3 - 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-39.9 - 69.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (15.4 - 87.8i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-85.5 + 71.7i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-47.1 + 267. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-180. + 312. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (10.5 + 8.82i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (383. + 139. i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-14.3 - 81.3i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 28.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-629. + 228. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (16.7 + 95.1i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-514. - 431. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (47.7 - 82.7i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (502. + 871. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-780. + 654. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (547. - 459. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-314. - 544. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.54e3 + 562. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.51162838422136340632327574349, −12.31744337159321012181242084595, −11.33634302321023456557602763369, −9.980748290669659046141576174546, −9.373279785134421843240107224814, −8.399455478072837157207432172878, −6.17104268389357440240493760104, −5.50322036004372573333866460578, −4.04232742300626587329328471940, −2.02421275323377128674591533329,
1.08894285642338017269256509005, 2.72326535823141370397255911521, 4.95777368792955027049660717155, 6.57078456243407618085027873972, 6.95706449333412137614778007356, 8.603947464204629005412633786205, 9.829644646700521709194223634947, 11.00082550603588710622332530960, 11.90125928173234505061352882904, 13.36141356348755790697300115694
