| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.810 + 1.77i)5-s + (0.142 − 0.989i)6-s + (0.439 − 0.129i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (1.87 + 0.549i)10-s + (−0.824 − 0.951i)11-s + (−0.654 − 0.755i)12-s + (−5.37 − 1.57i)13-s + (0.190 − 0.416i)14-s + (1.64 + 1.05i)15-s + (−0.959 + 0.281i)16-s + (−0.931 + 6.47i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.362 + 0.794i)5-s + (0.0580 − 0.404i)6-s + (0.166 − 0.0487i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.592 + 0.173i)10-s + (−0.248 − 0.287i)11-s + (−0.189 − 0.218i)12-s + (−1.49 − 0.437i)13-s + (0.0508 − 0.111i)14-s + (0.423 + 0.272i)15-s + (−0.239 + 0.0704i)16-s + (−0.225 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43192 - 0.588210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43192 - 0.588210i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-2.94 - 3.78i)T \) |
| good | 5 | \( 1 + (-0.810 - 1.77i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.439 + 0.129i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.824 + 0.951i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (5.37 + 1.57i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.931 - 6.47i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.301 + 2.09i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.720 - 5.01i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.0916 + 0.0589i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.384 - 0.842i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (4.08 + 8.95i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.69 + 3.66i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 2.50T + 47T^{2} \) |
| 53 | \( 1 + (-2.53 + 0.745i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-4.10 - 1.20i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (7.22 + 4.64i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.84 + 11.3i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.198 + 0.229i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.476 + 3.31i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.72 - 2.26i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.35 + 9.54i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-5.01 + 3.22i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.58 - 16.6i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.02750929483229331026116486347, −12.27288187020815260542033045261, −10.91718264946393984963475573170, −10.24282728247065663699930271867, −9.025881611766199366880797404859, −7.63644365496841096116595587075, −6.51203033371667758773752499217, −5.11626617812795187137912479449, −3.41111009847842034008797693747, −2.18531249379755773674055162769,
2.58704114076809872065577876083, 4.53929220037377712573234226574, 5.20008571102235240027344437115, 6.90217582750308945778905597709, 7.967129979963413220475039891623, 9.163465336284902334839597432456, 9.871505192243156160528541770079, 11.53786124743729571727447696683, 12.55803621111614115215377683607, 13.40483091332887802501175544359
