| L(s) = 1 | + (−0.472 + 1.33i)2-s + (1.64 − 0.541i)3-s + (−1.55 − 1.25i)4-s + (−1.99 + 1.00i)5-s + (−0.0547 + 2.44i)6-s + 3.00·7-s + (2.41 − 1.47i)8-s + (2.41 − 1.78i)9-s + (−0.399 − 3.13i)10-s + (3.97 + 2.88i)11-s + (−3.23 − 1.22i)12-s + (−1.00 − 1.37i)13-s + (−1.42 + 4.00i)14-s + (−2.73 + 2.73i)15-s + (0.828 + 3.91i)16-s + (−1.60 + 4.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)2-s + (0.949 − 0.312i)3-s + (−0.776 − 0.629i)4-s + (−0.892 + 0.450i)5-s + (−0.0223 + 0.999i)6-s + 1.13·7-s + (0.852 − 0.522i)8-s + (0.804 − 0.594i)9-s + (−0.126 − 0.991i)10-s + (1.19 + 0.870i)11-s + (−0.934 − 0.354i)12-s + (−0.277 − 0.381i)13-s + (−0.379 + 1.07i)14-s + (−0.707 + 0.707i)15-s + (0.207 + 0.978i)16-s + (−0.389 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20725 + 0.718880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20725 + 0.718880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.472 - 1.33i)T \) |
| 3 | \( 1 + (-1.64 + 0.541i)T \) |
| 5 | \( 1 + (1.99 - 1.00i)T \) |
| good | 7 | \( 1 - 3.00T + 7T^{2} \) |
| 11 | \( 1 + (-3.97 - 2.88i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.00 + 1.37i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.60 - 4.94i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.378 - 0.122i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.680 - 0.936i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-4.13 + 1.34i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (5.06 + 1.64i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.06 - 2.84i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.11 + 8.42i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + (10.9 - 3.56i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.93 + 9.02i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.303 + 0.220i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.49 + 4.71i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.185 - 0.570i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.83 + 8.71i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.41 - 8.83i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.30 - 0.749i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.583 - 0.189i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.36 + 3.24i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.41 - 1.10i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.99589755161493854346921933713, −10.81677654239132125915418842892, −9.740582868551652569888984209406, −8.688977493995576781373685790944, −8.006575980385573380453743049364, −7.30473631548195187723655532374, −6.40993364392416982963350319836, −4.64804620879523411033608822643, −3.81816448572930836404418899168, −1.68897194984999559932249706624,
1.39608314678931438270559061369, 3.02391944407428842420557325908, 4.18383984485752644788674027918, 4.85122908314157892002076710459, 7.23742889598307940117926896992, 8.142201463987172834149896038523, 8.847282387911858508467773998594, 9.448065903210878001087448562460, 10.86332136330299758218778569229, 11.49769830801072855076222924984
