| L(s) = 1 | + (0.670 + 1.59i)3-s + (−2.14 + 0.633i)5-s + (0.907 + 0.907i)7-s + (−2.10 + 2.14i)9-s + (−1.03 + 0.337i)11-s + (2.26 + 4.44i)13-s + (−2.45 − 2.99i)15-s + (−2.19 + 0.347i)17-s + (−2.39 + 3.30i)19-s + (−0.841 + 2.05i)21-s + (2.13 − 4.18i)23-s + (4.19 − 2.71i)25-s + (−4.82 − 1.91i)27-s + (−0.981 + 0.712i)29-s + (0.992 + 0.720i)31-s + ⋯ |
| L(s) = 1 | + (0.387 + 0.921i)3-s + (−0.959 + 0.283i)5-s + (0.343 + 0.343i)7-s + (−0.700 + 0.713i)9-s + (−0.313 + 0.101i)11-s + (0.628 + 1.23i)13-s + (−0.632 − 0.774i)15-s + (−0.532 + 0.0843i)17-s + (−0.550 + 0.757i)19-s + (−0.183 + 0.449i)21-s + (0.445 − 0.873i)23-s + (0.839 − 0.543i)25-s + (−0.929 − 0.369i)27-s + (−0.182 + 0.132i)29-s + (0.178 + 0.129i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.627901 + 0.954125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627901 + 0.954125i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.670 - 1.59i)T \) |
| 5 | \( 1 + (2.14 - 0.633i)T \) |
| good | 7 | \( 1 + (-0.907 - 0.907i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.03 - 0.337i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.26 - 4.44i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.19 - 0.347i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (2.39 - 3.30i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.13 + 4.18i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (0.981 - 0.712i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.992 - 0.720i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.47 + 3.29i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-8.57 - 2.78i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.48 + 1.48i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.0645 - 0.407i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-13.7 - 2.18i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (3.76 - 11.6i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.344 - 1.05i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.22 + 7.74i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-6.04 - 8.32i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.93 - 3.02i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (5.56 + 7.66i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.79 + 17.6i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (4.10 + 12.6i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (15.3 + 2.42i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.74274217658106455870281955100, −11.08110594677214376096772075777, −10.31213202734643800041825939607, −8.995438607689686202445808922613, −8.475147036059510918218860792005, −7.37869984169740006416681060142, −6.05703398733184144367436601599, −4.57677015593277589004513955456, −3.92909551969227069476521849657, −2.47242875031311437277959364132,
0.839810208650512861875347652706, 2.78002743060574680824462473704, 4.04412592277699080253289746808, 5.47069138792319682991193222848, 6.78586983440914806155844263655, 7.77036206803383206913955702738, 8.275033622491548216661747238732, 9.297048068641461252251031998792, 10.90313365063401927846384883430, 11.37428795069266384722615882190
