| L(s) = 1 | + (1.63 − 1.41i)3-s + (−0.135 + 0.0195i)5-s + (0.888 − 1.94i)7-s + (0.235 − 1.63i)9-s + (1.23 − 4.21i)11-s + (−1.14 + 0.522i)13-s + (−0.193 + 0.223i)15-s + (−0.710 − 0.456i)17-s + (−1.64 − 2.56i)19-s + (−1.30 − 4.42i)21-s + (2.25 + 4.23i)23-s + (−4.77 + 1.40i)25-s + (1.56 + 2.44i)27-s + (3.45 − 5.38i)29-s + (2.47 − 2.85i)31-s + ⋯ |
| L(s) = 1 | + (0.941 − 0.815i)3-s + (−0.0607 + 0.00873i)5-s + (0.335 − 0.735i)7-s + (0.0785 − 0.546i)9-s + (0.373 − 1.27i)11-s + (−0.317 + 0.144i)13-s + (−0.0500 + 0.0577i)15-s + (−0.172 − 0.110i)17-s + (−0.377 − 0.588i)19-s + (−0.283 − 0.966i)21-s + (0.471 + 0.882i)23-s + (−0.955 + 0.280i)25-s + (0.301 + 0.469i)27-s + (0.642 − 0.999i)29-s + (0.444 − 0.512i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0715 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0715 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50516 - 1.40104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50516 - 1.40104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-2.25 - 4.23i)T \) |
| good | 3 | \( 1 + (-1.63 + 1.41i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.135 - 0.0195i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.888 + 1.94i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 4.21i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.522i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.710 + 0.456i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.64 + 2.56i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.45 + 5.38i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.47 + 2.85i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-4.39 - 0.631i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.715 - 4.97i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.55 + 4.81i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + (4.41 + 2.01i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-6.76 + 3.08i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.47 - 5.60i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.447 - 1.52i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (8.44 - 2.47i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (2.17 - 1.39i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.635 + 1.39i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (16.4 + 2.36i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.52 - 5.22i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.25 - 15.6i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.06599435199323384596661759955, −9.133451950080294663760807749624, −8.296225195253844064918205530451, −7.70526270702393530307661644405, −6.88974702123591336358962335187, −5.90091542133114946681356109069, −4.52529996490042889564153227977, −3.43845348478528007294025464765, −2.37209067621467776195557402167, −1.01413494197501676245568627147,
1.97777894860014412657548399135, 2.99080979143932979526894316058, 4.21315433063234219426748148127, 4.84409100187075477286650512981, 6.15359037645113406989703184860, 7.24198280318197971015228519661, 8.328533826560310158784829673171, 8.824274693596112627225229853112, 9.774275755068907833641606282407, 10.20889554829039143403805552121
