| L(s) = 1 | + (−0.553 + 0.638i)2-s + (2.80 − 1.80i)3-s + (0.183 + 1.27i)4-s + (0.415 + 0.909i)5-s + (−0.401 + 2.79i)6-s + (−3.04 + 0.893i)7-s + (−2.33 − 1.50i)8-s + (3.38 − 7.41i)9-s + (−0.810 − 0.238i)10-s + (0.590 + 0.681i)11-s + (2.81 + 3.24i)12-s + (−3.41 − 1.00i)13-s + (1.11 − 2.43i)14-s + (2.80 + 1.80i)15-s + (−0.217 + 0.0637i)16-s + (0.257 − 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.391 + 0.451i)2-s + (1.62 − 1.04i)3-s + (0.0915 + 0.636i)4-s + (0.185 + 0.406i)5-s + (−0.163 + 1.14i)6-s + (−1.14 + 0.337i)7-s + (−0.825 − 0.530i)8-s + (1.12 − 2.47i)9-s + (−0.256 − 0.0752i)10-s + (0.178 + 0.205i)11-s + (0.812 + 0.937i)12-s + (−0.948 − 0.278i)13-s + (0.297 − 0.651i)14-s + (0.725 + 0.466i)15-s + (−0.0542 + 0.0159i)16-s + (0.0623 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25424 + 0.157298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25424 + 0.157298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-3.69 + 3.05i)T \) |
| good | 2 | \( 1 + (0.553 - 0.638i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-2.80 + 1.80i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (3.04 - 0.893i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.590 - 0.681i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.41 + 1.00i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.257 + 1.78i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.483 - 3.36i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.318 - 2.21i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.70 + 2.38i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (4.29 - 9.39i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.56 - 3.41i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.88 - 1.21i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + (3.23 - 0.949i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-4.90 - 1.43i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 2.33i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-5.90 + 6.81i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.03 + 4.65i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.0262 - 0.182i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-8.14 - 2.39i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.04 + 8.86i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (5.81 - 3.73i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.51 - 14.2i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.55406323312242021777384977476, −12.68236720634458227116027853774, −12.15518191141323334393185837154, −9.803654017187844419558269926337, −9.151276564055096031645260346747, −8.084410592900680528383270844042, −7.15767588702429290519163799873, −6.49037011994246021573420231430, −3.43688316545227890334031045307, −2.59768172608463829038917741999,
2.36813447233157545690487136152, 3.68764553123296114086758268349, 5.21827365051603930339414490702, 7.16075535259731217262601663662, 8.769110679420603855076278382211, 9.394062907533948167305175696619, 9.988933144069210881049517902394, 10.95137399620430678008642239298, 12.72524353695492718541218736468, 13.75757935656009769022815495611
