| L(s) = 1 | + (−0.117 + 1.40i)2-s + (−2.14 − 2.28i)3-s + (−1.97 − 0.331i)4-s + (0.173 + 0.383i)5-s + (3.47 − 2.74i)6-s + (−2.08 + 1.63i)7-s + (0.698 − 2.74i)8-s + (−0.445 + 6.79i)9-s + (−0.561 + 0.199i)10-s + (−1.02 − 1.64i)11-s + (3.46 + 5.21i)12-s + (0.154 + 1.56i)13-s + (−2.05 − 3.12i)14-s + (0.504 − 1.21i)15-s + (3.78 + 1.30i)16-s + (0.0405 + 0.00533i)17-s + ⋯ |
| L(s) = 1 | + (−0.0830 + 0.996i)2-s + (−1.23 − 1.31i)3-s + (−0.986 − 0.165i)4-s + (0.0777 + 0.171i)5-s + (1.41 − 1.12i)6-s + (−0.786 + 0.617i)7-s + (0.246 − 0.969i)8-s + (−0.148 + 2.26i)9-s + (−0.177 + 0.0632i)10-s + (−0.307 − 0.494i)11-s + (1.00 + 1.50i)12-s + (0.0428 + 0.435i)13-s + (−0.550 − 0.835i)14-s + (0.130 − 0.314i)15-s + (0.945 + 0.326i)16-s + (0.00983 + 0.00129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.584376 - 0.0390341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.584376 - 0.0390341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.117 - 1.40i)T \) |
| 7 | \( 1 + (2.08 - 1.63i)T \) |
| good | 3 | \( 1 + (2.14 + 2.28i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.383i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.02 + 1.64i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (-0.154 - 1.56i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-0.0405 - 0.00533i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (1.03 - 6.29i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (5.40 + 6.16i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.61 - 6.76i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (4.31 - 1.15i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-4.11 + 10.9i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (1.12 + 5.66i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-9.81 + 2.97i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-5.35 + 6.97i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-4.54 - 7.30i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-3.44 + 4.81i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-4.20 + 0.980i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-5.82 - 6.21i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-2.74 - 4.10i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (5.33 - 2.63i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (0.0103 + 0.0785i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 5.99i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (2.23 + 6.58i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (9.46 - 9.46i)T - 97iT^{2} \) |
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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
−10.27694922698476832525735117281, −8.917794670696016515563796504975, −8.225960991339045022453630173027, −7.21548305334003333705244098191, −6.63360722307236431003833042677, −5.82900816796729859829258430113, −5.52717438823915745807312282506, −4.04626807260891286434786986375, −2.23444925252147511363034949001, −0.54755353700179653248086694032,
0.75994325433620467542638988531, 2.88100743561963115184937982621, 3.94705127089706557608658890005, 4.60923671193822969084237281692, 5.44820052018411345463659383270, 6.38238443255579758981387472956, 7.68376592764388736846348277330, 9.042784151998177548067085078514, 9.819957894822040935291960312179, 10.00463303525525292379321280415
