| L(s) = 1 | + (0.959 + 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.0651 + 0.453i)5-s + (0.841 − 0.540i)6-s + (−0.134 − 0.295i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.190 + 0.416i)10-s + (−2.38 + 0.701i)11-s + (0.959 − 0.281i)12-s + (0.564 − 1.23i)13-s + (−0.0462 − 0.321i)14-s + (0.299 + 0.346i)15-s + (0.415 + 0.909i)16-s + (−2.26 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.378 − 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0291 + 0.202i)5-s + (0.343 − 0.220i)6-s + (−0.0509 − 0.111i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.0601 + 0.131i)10-s + (−0.719 + 0.211i)11-s + (0.276 − 0.0813i)12-s + (0.156 − 0.342i)13-s + (−0.0123 − 0.0858i)14-s + (0.0774 + 0.0893i)15-s + (0.103 + 0.227i)16-s + (−0.550 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64765 + 0.0221112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64765 + 0.0221112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (4.35 - 2.00i)T \) |
| good | 5 | \( 1 + (0.0651 - 0.453i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.134 + 0.295i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.38 - 0.701i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.564 + 1.23i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.26 - 1.45i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.43 + 1.56i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.83 - 1.17i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.61 + 1.86i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0683 + 0.475i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.120 - 0.838i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.76 + 6.65i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 + (-3.62 - 7.93i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.52 + 9.89i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-7.89 - 9.11i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (10.1 + 2.97i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-15.7 - 4.62i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.89 - 6.35i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (4.45 - 9.75i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.15 + 8.00i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.74 + 7.78i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.0319 + 0.221i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−13.16869648491521545838283789784, −12.55538218199183323521418130206, −11.28743602867960557798794017649, −10.29297011654148769958862821478, −8.783340976292928377836286459246, −7.68767459575263050383799525693, −6.70486749085688156525665364724, −5.43773865802009346703657157176, −3.90043981022277574650530208845, −2.39725447086874115817248376757,
2.46115382359368074700392390996, 3.98125946578546249138041784781, 5.12893402197946579554911042519, 6.45677070662978907198268362054, 7.946805987342798098087675154956, 9.064594894979581372436338074545, 10.28209479343438458359229527919, 11.14104949179606379915222884452, 12.33682275752013643127784174976, 13.22271022879247047530606205453
