| L(s) = 1 | + (−0.977 + 0.212i)2-s + (−1.72 + 2.30i)3-s + (0.909 − 0.415i)4-s + (0.267 − 2.22i)5-s + (1.19 − 2.61i)6-s + (−3.05 + 0.218i)7-s + (−0.800 + 0.599i)8-s + (−1.48 − 5.07i)9-s + (0.210 + 2.22i)10-s + (−0.582 − 0.906i)11-s + (−0.611 + 2.81i)12-s + (0.0976 − 1.36i)13-s + (2.93 − 0.863i)14-s + (4.65 + 4.44i)15-s + (0.654 − 0.755i)16-s + (5.82 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.150i)2-s + (−0.995 + 1.33i)3-s + (0.454 − 0.207i)4-s + (0.119 − 0.992i)5-s + (0.488 − 1.06i)6-s + (−1.15 + 0.0826i)7-s + (−0.283 + 0.211i)8-s + (−0.496 − 1.69i)9-s + (0.0665 + 0.703i)10-s + (−0.175 − 0.273i)11-s + (−0.176 + 0.811i)12-s + (0.0270 − 0.378i)13-s + (0.785 − 0.230i)14-s + (1.20 + 1.14i)15-s + (0.163 − 0.188i)16-s + (1.41 − 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247537 - 0.193784i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247537 - 0.193784i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (-0.267 + 2.22i)T \) |
| 23 | \( 1 + (2.80 + 3.89i)T \) |
| good | 3 | \( 1 + (1.72 - 2.30i)T + (-0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (3.05 - 0.218i)T + (6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.582 + 0.906i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.0976 + 1.36i)T + (-12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-5.82 + 2.17i)T + (12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (2.12 + 4.66i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-6.11 - 2.79i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.518 - 3.60i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (7.48 + 4.08i)T + (20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (10.3 + 3.03i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (2.52 + 1.88i)T + (12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-1.06 + 1.06i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.318 - 4.45i)T + (-52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-4.89 + 4.23i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (7.99 + 1.15i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (2.89 + 13.3i)T + (-60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-2.34 - 1.50i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.79 - 12.8i)T + (-55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-0.149 - 0.172i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.51 + 6.43i)T + (-44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-1.55 - 10.8i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 - 5.50i)T + (-52.4 + 81.6i)T^{2} \) |
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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
−11.96783165933485536461211488649, −10.63300462383536639779501815082, −10.09301317427100032857549354679, −9.289419753389927639260856073288, −8.451386840548898717416105223623, −6.73964500419409143681920484232, −5.66543115133546279241860860571, −4.87400134759766489391906476667, −3.34720436437685982415823268353, −0.35477847175839665479039465415,
1.74379240730917905615303335243, 3.33812003974425539162934520437, 5.84643302794090320813479062641, 6.44341277379988184110464909412, 7.27269909271943251535189655036, 8.143054209258341815319010062254, 9.986951380547976748866787131881, 10.29374763586044943585360859249, 11.67453586239395676974553257578, 12.12402625065337330972837256219
