L(s) = 1 | + (2.57 + 2.57i)5-s + 3.74·7-s + (−3.64 + 3.64i)11-s + (−4.64 − 4.64i)13-s + 0.913i·17-s + (−1.41 + 1.41i)19-s + 6i·23-s + 8.29i·25-s + (−1.66 + 1.66i)29-s + 6.57i·31-s + (9.64 + 9.64i)35-s + (−1 + i)37-s − 0.913·41-s + (3.74 + 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (1.15 + 1.15i)5-s + 1.41·7-s + (−1.09 + 1.09i)11-s + (−1.28 − 1.28i)13-s + 0.221i·17-s + (−0.324 + 0.324i)19-s + 1.25i·23-s + 1.65i·25-s + (−0.309 + 0.309i)29-s + 1.18i·31-s + (1.63 + 1.63i)35-s + (−0.164 + 0.164i)37-s − 0.142·41-s + (0.570 + 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.022304485\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.022304485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.57 - 2.57i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + (3.64 - 3.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.64 + 4.64i)T + 13iT^{2} \) |
| 17 | \( 1 - 0.913iT - 17T^{2} \) |
| 19 | \( 1 + (1.41 - 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (1.66 - 1.66i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.57iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.913T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (3.49 + 3.49i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.29 + 1.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.291 - 0.291i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.32 - 3.32i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 8.58iT - 73T^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (4.93 + 4.93i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.438919917205187162814839918981, −8.280335855949084207745817611564, −7.46314907205754856921754489130, −7.23363908931494355673378084870, −5.93657803145602315244150105265, −5.26368119152462175845754143482, −4.74480844087056492374612637277, −3.20443046223421931147318411012, −2.35402857228229840295596833234, −1.69385847957087126049316656400,
0.63754524525460717533649850051, 1.99017875440674645688320945468, 2.45178195391633857090456911867, 4.32233427649618824712410618781, 4.85471221686446575088691927347, 5.46344730452389759881215186134, 6.24512296934969818220901408953, 7.41961290887867617711784029956, 8.142913931456640551169192162186, 8.848359610709895039932408526851