| L(s) = 1 | + (−1.87 − 0.687i)2-s + (−0.477 + 2.96i)3-s + (3.05 + 2.58i)4-s + (−0.175 − 0.996i)5-s + (2.93 − 5.23i)6-s + (−4.47 − 3.75i)7-s + (−3.95 − 6.95i)8-s + (−8.54 − 2.83i)9-s + (−0.355 + 1.99i)10-s + (1.81 − 10.3i)11-s + (−9.11 + 7.80i)12-s + (−2.68 + 7.37i)13-s + (5.81 + 10.1i)14-s + (3.03 − 0.0442i)15-s + (2.64 + 15.7i)16-s + (27.5 − 15.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 − 0.343i)2-s + (−0.159 + 0.987i)3-s + (0.763 + 0.645i)4-s + (−0.0351 − 0.199i)5-s + (0.489 − 0.872i)6-s + (−0.638 − 0.535i)7-s + (−0.494 − 0.869i)8-s + (−0.949 − 0.314i)9-s + (−0.0355 + 0.199i)10-s + (0.165 − 0.937i)11-s + (−0.759 + 0.650i)12-s + (−0.206 + 0.567i)13-s + (0.415 + 0.722i)14-s + (0.202 − 0.00294i)15-s + (0.165 + 0.986i)16-s + (1.61 − 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.536859 - 0.384263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.536859 - 0.384263i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 + 0.687i)T \) |
| 3 | \( 1 + (0.477 - 2.96i)T \) |
| good | 5 | \( 1 + (0.175 + 0.996i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (4.47 + 3.75i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 10.3i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (2.68 - 7.37i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-27.5 + 15.8i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4.05 - 2.34i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (24.0 + 28.7i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (12.2 - 4.46i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (13.0 - 10.9i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-45.8 + 26.4i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.93 + 13.5i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (38.8 + 6.85i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-53.2 + 63.4i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 39.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-8.89 - 50.4i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (10.3 - 12.3i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-9.18 + 25.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (30.4 - 17.5i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-25.8 + 44.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (89.2 - 32.4i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (119. - 43.3i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-81.2 - 46.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (9.94 - 56.3i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.68255968913650777476974102858, −10.66218337750043684746536666249, −9.963090791979037514426960650969, −9.172940547881986011749835134883, −8.227018546189662837591896177851, −6.92960783489750777814141772550, −5.69425217245375856876723479328, −4.01332788676354843992206423915, −2.96906758220783719363814765661, −0.52744347002067592466288676810,
1.47031808093010498310384614034, 2.95835965417981534463464504106, 5.53843749223191486057775310041, 6.27369113257088064085033800578, 7.46509723523124256680956753762, 7.997837445859943245057186959069, 9.365961207615791573449280609710, 10.10754865795796405418409230780, 11.34870031802948223314769369360, 12.24613427933505706360921768574
