| L(s) = 1 | + (0.104 + 0.994i)2-s + (−1.03 + 0.749i)3-s + (−0.978 + 0.207i)4-s + (1.51 + 1.68i)5-s + (−0.853 − 0.947i)6-s + (−1.54 + 0.686i)7-s + (−0.309 − 0.951i)8-s + (−0.424 + 1.30i)9-s + (−1.51 + 1.68i)10-s − 0.786·11-s + (0.853 − 0.947i)12-s + (0.135 − 0.233i)13-s + (−0.843 − 1.46i)14-s + (−2.83 − 0.601i)15-s + (0.913 − 0.406i)16-s + (6.23 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.595 + 0.432i)3-s + (−0.489 + 0.103i)4-s + (0.678 + 0.754i)5-s + (−0.348 − 0.386i)6-s + (−0.582 + 0.259i)7-s + (−0.109 − 0.336i)8-s + (−0.141 + 0.435i)9-s + (−0.480 + 0.533i)10-s − 0.237·11-s + (0.246 − 0.273i)12-s + (0.0374 − 0.0648i)13-s + (−0.225 − 0.390i)14-s + (−0.730 − 0.155i)15-s + (0.228 − 0.101i)16-s + (1.51 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.461904 + 0.776397i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.461904 + 0.776397i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (-2.64 + 7.34i)T \) |
| good | 3 | \( 1 + (1.03 - 0.749i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.51 - 1.68i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (1.54 - 0.686i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + 0.786T + 11T^{2} \) |
| 13 | \( 1 + (-0.135 + 0.233i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-6.23 + 1.32i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-3.66 - 1.63i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.89 + 5.83i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.46 - 2.54i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.480 - 4.57i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-5.36 - 3.89i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.19 + 3.77i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.91 + 0.406i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (3.91 + 6.78i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.25 + 6.93i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.495 - 4.71i)T + (-57.7 + 12.2i)T^{2} \) |
| 67 | \( 1 + (-7.59 - 8.43i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (6.28 - 6.98i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (7.48 - 8.31i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.836 + 0.177i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (1.27 + 12.1i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 7.72i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.974 + 9.27i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−14.05051883827702569853042937306, −12.90230076894727273693332329859, −11.72370423068404762146027373354, −10.32866718221522763006856013899, −9.892195342042546459407561941863, −8.343276040529191469367470951576, −6.98820661448191112680208512666, −5.92029838574009609541540508012, −5.04699307879688824618586982465, −3.06717318269651761805549497129,
1.17227763534508656830702540979, 3.36599131838851260890336404602, 5.20886996879690388367945187562, 6.11025779943557636884926508148, 7.68200526601307613156700866187, 9.310415837220588187200063106271, 9.840965208567768946665310024888, 11.26619192846475101589543607985, 12.15042039318105421613635460154, 12.98988772904920631804054133355
