| L(s) = 1 | + (1.02 − 0.724i)2-s + (−1.71 + 0.207i)3-s + (−0.141 + 0.397i)4-s + (2.35 − 2.51i)5-s + (−1.61 + 1.45i)6-s + (1.69 − 3.27i)7-s + (0.820 + 2.92i)8-s + (2.91 − 0.712i)9-s + (0.589 − 4.28i)10-s + (−1.22 − 0.744i)11-s + (0.160 − 0.713i)12-s + (−1.80 + 4.16i)13-s + (−0.630 − 4.58i)14-s + (−3.52 + 4.81i)15-s + (2.30 + 1.87i)16-s + (−0.579 − 0.952i)17-s + ⋯ |
| L(s) = 1 | + (0.725 − 0.512i)2-s + (−0.992 + 0.119i)3-s + (−0.0706 + 0.198i)4-s + (1.05 − 1.12i)5-s + (−0.659 + 0.595i)6-s + (0.640 − 1.23i)7-s + (0.290 + 1.03i)8-s + (0.971 − 0.237i)9-s + (0.186 − 1.35i)10-s + (−0.369 − 0.224i)11-s + (0.0463 − 0.205i)12-s + (−0.501 + 1.15i)13-s + (−0.168 − 1.22i)14-s + (−0.909 + 1.24i)15-s + (0.577 + 0.469i)16-s + (−0.140 − 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22186 - 0.560303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22186 - 0.560303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 - 0.207i)T \) |
| 47 | \( 1 + (2.60 + 6.34i)T \) |
| good | 2 | \( 1 + (-1.02 + 0.724i)T + (0.669 - 1.88i)T^{2} \) |
| 5 | \( 1 + (-2.35 + 2.51i)T + (-0.341 - 4.98i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 3.27i)T + (-4.03 - 5.71i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.744i)T + (5.06 + 9.76i)T^{2} \) |
| 13 | \( 1 + (1.80 - 4.16i)T + (-8.87 - 9.50i)T^{2} \) |
| 17 | \( 1 + (0.579 + 0.952i)T + (-7.82 + 15.0i)T^{2} \) |
| 19 | \( 1 + (4.81 - 4.49i)T + (1.29 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.610 + 0.864i)T + (-7.70 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-1.78 + 0.774i)T + (19.7 - 21.1i)T^{2} \) |
| 31 | \( 1 + (3.85 - 4.73i)T + (-6.30 - 30.3i)T^{2} \) |
| 37 | \( 1 + (-5.58 - 0.767i)T + (35.6 + 9.98i)T^{2} \) |
| 41 | \( 1 + (-4.96 - 1.39i)T + (35.0 + 21.3i)T^{2} \) |
| 43 | \( 1 + (-6.04 - 2.15i)T + (33.3 + 27.1i)T^{2} \) |
| 53 | \( 1 + (1.59 - 5.69i)T + (-45.2 - 27.5i)T^{2} \) |
| 59 | \( 1 + (7.11 - 2.52i)T + (45.7 - 37.2i)T^{2} \) |
| 61 | \( 1 + (9.37 - 1.28i)T + (58.7 - 16.4i)T^{2} \) |
| 67 | \( 1 + (-4.49 + 2.32i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (11.5 + 8.16i)T + (23.7 + 66.9i)T^{2} \) |
| 73 | \( 1 + (-0.761 + 0.158i)T + (66.9 - 29.0i)T^{2} \) |
| 79 | \( 1 + (0.366 - 5.35i)T + (-78.2 - 10.7i)T^{2} \) |
| 83 | \( 1 + (-4.53 + 7.46i)T + (-38.1 - 73.6i)T^{2} \) |
| 89 | \( 1 + (4.32 + 4.03i)T + (6.07 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-14.0 + 11.4i)T + (19.7 - 94.9i)T^{2} \) |
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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
−12.89126381657153377848217482897, −12.20347929250904049423495171892, −11.10513155552801907475478378398, −10.28204161758277653451785018887, −9.003878020727839179097763237253, −7.59803265546640653074177360630, −6.04352271689756462656097806158, −4.78731550276558838001130957802, −4.30335611016560343204379344766, −1.68498975119399406576480977570,
2.36914541938674579668181936502, 4.79181291179168134023172961466, 5.71987260136519145263887287178, 6.27857146751605921252649157585, 7.49233837624609297351987921380, 9.414240050757921683746793039480, 10.44558060130441756152431360338, 11.12574036871997783530590566515, 12.61140467681069739926536470096, 13.18306966141762381622762491400
