| L(s) = 1 | + (1.38 + 0.263i)2-s + (−0.331 − 2.19i)3-s + (1.86 + 0.731i)4-s + (2.18 + 0.855i)5-s + (0.118 − 3.14i)6-s + (0.239 − 2.63i)7-s + (2.39 + 1.50i)8-s + (−1.86 + 0.574i)9-s + (2.80 + 1.76i)10-s + (−0.946 + 3.06i)11-s + (0.992 − 4.33i)12-s + (−6.29 + 1.43i)13-s + (1.02 − 3.59i)14-s + (1.15 − 5.08i)15-s + (2.92 + 2.72i)16-s + (1.11 − 0.763i)17-s + ⋯ |
| L(s) = 1 | + (0.982 + 0.186i)2-s + (−0.191 − 1.27i)3-s + (0.930 + 0.365i)4-s + (0.975 + 0.382i)5-s + (0.0483 − 1.28i)6-s + (0.0906 − 0.995i)7-s + (0.846 + 0.532i)8-s + (−0.620 + 0.191i)9-s + (0.886 + 0.557i)10-s + (−0.285 + 0.925i)11-s + (0.286 − 1.25i)12-s + (−1.74 + 0.398i)13-s + (0.274 − 0.961i)14-s + (0.299 − 1.31i)15-s + (0.732 + 0.680i)16-s + (0.271 − 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43330 - 0.889004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43330 - 0.889004i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.263i)T \) |
| 7 | \( 1 + (-0.239 + 2.63i)T \) |
| good | 3 | \( 1 + (0.331 + 2.19i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-2.18 - 0.855i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.946 - 3.06i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (6.29 - 1.43i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 0.763i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.56 + 1.48i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.865 + 0.589i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (4.05 - 8.42i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.06 + 5.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.51 - 0.188i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 1.78i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.45 - 1.15i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (5.07 - 4.70i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-13.1 - 0.984i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-3.35 + 1.31i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (8.59 - 0.644i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (5.87 + 3.39i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.1 - 5.36i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.95 - 6.44i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.16 - 5.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.04 + 1.37i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-9.55 + 2.94i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 0.651T + 97T^{2} \) |
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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
−11.57522208656885672959139184344, −10.37599760951600709299132972854, −9.665403468986001038199947077235, −7.68002242229817855212731777785, −7.23221375424736678398392538920, −6.63356971718654045759505072994, −5.46731039519872229138492385810, −4.45700626551231991406312484606, −2.67721335216199238338600838938, −1.71770018524755465507861836965,
2.20667472758897531879961071114, 3.36095910237470092605143779411, 4.76979376051271046272274376107, 5.43257711560713923093190315463, 5.94639002043947047638347401430, 7.60381262458533087487585067563, 9.016170342592323537081424126181, 9.932395596718086915367570751535, 10.31462041425646747003711863880, 11.55774511480670406387524352209
