L(s) = 1 | + 2.22·2-s + (0.274 − 0.475i)3-s + 2.92·4-s + (−2.11 + 3.65i)5-s + (0.610 − 1.05i)6-s + 2.06·8-s + (1.34 + 2.33i)9-s + (−4.68 + 8.11i)10-s + (0.274 − 0.475i)11-s + (0.804 − 1.39i)12-s + (2.95 + 2.06i)13-s + (1.15 + 2.00i)15-s − 1.28·16-s + 2.37·17-s + (2.99 + 5.18i)18-s + (−1.80 − 3.12i)19-s + ⋯ |
L(s) = 1 | + 1.56·2-s + (0.158 − 0.274i)3-s + 1.46·4-s + (−0.943 + 1.63i)5-s + (0.249 − 0.431i)6-s + 0.728·8-s + (0.449 + 0.778i)9-s + (−1.48 + 2.56i)10-s + (0.0828 − 0.143i)11-s + (0.232 − 0.402i)12-s + (0.820 + 0.571i)13-s + (0.299 + 0.518i)15-s − 0.320·16-s + 0.576·17-s + (0.705 + 1.22i)18-s + (−0.414 − 0.717i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.83160 + 1.47544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.83160 + 1.47544i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.95 - 2.06i)T \) |
good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 3 | \( 1 + (-0.274 + 0.475i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.11 - 3.65i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.274 + 0.475i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 + (1.80 + 3.12i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 + (-1.79 - 3.11i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.57 + 4.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.329T + 37T^{2} \) |
| 41 | \( 1 + (3.14 + 5.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.61 - 2.78i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.10 + 7.10i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.32 + 2.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 + (-0.304 - 0.527i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.18 - 8.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.45 - 4.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.00 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.73T + 83T^{2} \) |
| 89 | \( 1 - 7.46T + 89T^{2} \) |
| 97 | \( 1 + (3.42 - 5.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.04562712849744609056044144820, −10.39060177408648676795235007002, −8.796960442082211827297448500289, −7.58628553783228862342960919642, −6.94482806298540982004961806382, −6.32321218850097244199177795121, −5.04577092326313382428623035128, −3.98335188131078060781162830396, −3.26975796979924572336048408292, −2.28921532457010512146987464176,
1.17500157539383653375110063634, 3.31594759062412008284421707768, 3.95157707780774191006152386089, 4.74869132031029046810771249306, 5.51019224297335261803323495448, 6.57176683976862850323005211521, 7.78839524075200347459422889034, 8.680139881070326756800067394537, 9.438686600668517855955583013232, 10.76919389776619675972259476618