L(s) = 1 | + (−1.95 − 3.39i)5-s + (1.96 − 1.77i)7-s + (3.19 + 1.84i)11-s + 0.554i·13-s + (2.91 − 5.05i)17-s + (4.62 − 2.66i)19-s + (1.96 − 1.13i)23-s + (−5.16 + 8.94i)25-s − 4.08i·29-s + (7.00 + 4.04i)31-s + (−9.85 − 3.18i)35-s + (3.89 + 6.75i)37-s − 7.18·41-s + 1.50·43-s + (1.41 + 2.44i)47-s + ⋯ |
L(s) = 1 | + (−0.875 − 1.51i)5-s + (0.742 − 0.670i)7-s + (0.964 + 0.556i)11-s + 0.153i·13-s + (0.707 − 1.22i)17-s + (1.06 − 0.612i)19-s + (0.410 − 0.237i)23-s + (−1.03 + 1.78i)25-s − 0.758i·29-s + (1.25 + 0.726i)31-s + (−1.66 − 0.538i)35-s + (0.640 + 1.11i)37-s − 1.12·41-s + 0.229·43-s + (0.206 + 0.357i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.808409419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808409419\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.96 + 1.77i)T \) |
good | 5 | \( 1 + (1.95 + 3.39i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.19 - 1.84i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.554iT - 13T^{2} \) |
| 17 | \( 1 + (-2.91 + 5.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.62 + 2.66i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.96 + 1.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.08iT - 29T^{2} \) |
| 31 | \( 1 + (-7.00 - 4.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0415 + 0.0239i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.45 + 7.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.03 - 3.48i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.587 - 1.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.71iT - 71T^{2} \) |
| 73 | \( 1 + (3.52 + 2.03i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.69T + 83T^{2} \) |
| 89 | \( 1 + (2.71 + 4.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−8.771677073680540372159307545973, −8.017164263520897113994534872064, −7.44111108140560863648075285749, −6.67094762417171455055263342906, −5.26086117880841553224259863447, −4.72549783086934700340233887456, −4.21689340429718345809085402125, −3.10636546998227733410853476650, −1.40132108819806347676460463265, −0.76374315374517573361042705057,
1.33798265588698071817172361597, 2.67888033484685182805190111929, 3.49202229973856933187096576759, 4.13005992486789841520316920265, 5.47669255570930102332612229102, 6.14257981259873991138801071676, 6.96378701108215867307356597649, 7.78224617755438050030829370652, 8.246165963720707571792679606308, 9.165141930432984236766927036613