| L(s) = 1 | + (−0.262 − 0.809i)2-s + (−0.809 + 0.587i)3-s + (1.03 − 0.749i)4-s + (2.19 + 0.407i)5-s + (0.688 + 0.500i)6-s + 7-s + (−2.25 − 1.63i)8-s + (0.309 − 0.951i)9-s + (−0.248 − 1.88i)10-s + (−1.56 − 4.80i)11-s + (−0.394 + 1.21i)12-s + (−0.139 + 0.428i)13-s + (−0.262 − 0.809i)14-s + (−2.01 + 0.962i)15-s + (0.0552 − 0.169i)16-s + (2.91 + 2.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.185 − 0.572i)2-s + (−0.467 + 0.339i)3-s + (0.516 − 0.374i)4-s + (0.983 + 0.182i)5-s + (0.281 + 0.204i)6-s + 0.377·7-s + (−0.797 − 0.579i)8-s + (0.103 − 0.317i)9-s + (−0.0784 − 0.596i)10-s + (−0.470 − 1.44i)11-s + (−0.113 + 0.350i)12-s + (−0.0385 + 0.118i)13-s + (−0.0702 − 0.216i)14-s + (−0.521 + 0.248i)15-s + (0.0138 − 0.0424i)16-s + (0.706 + 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23322 - 0.886422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23322 - 0.886422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-2.19 - 0.407i)T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + (0.262 + 0.809i)T + (-1.61 + 1.17i)T^{2} \) |
| 11 | \( 1 + (1.56 + 4.80i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.139 - 0.428i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.91 - 2.11i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.70 - 1.96i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.551 + 1.69i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.874 + 0.635i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.36 + 1.71i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.176 + 0.543i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.73 + 11.4i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 + (8.02 - 5.82i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.654 - 0.475i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.60 + 8.01i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.739 - 2.27i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 7.98i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (4.54 - 3.29i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.85 - 11.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.90 - 1.38i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.08 + 2.96i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.73 - 14.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.36 - 1.71i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.75884519703677204930451968413, −10.02429911088573735654342960322, −9.269414677493465975754394303475, −8.143384692745678796053342149832, −6.80212394590505220957328642597, −5.78784221389158927500414696502, −5.45463642286968568228940211995, −3.63394855120813328861196177507, −2.48078367332796625636985215544, −1.09545611951571850860051351888,
1.70077118952805268797073047311, 2.84392553395533982714388459269, 4.83968165958510672625175717139, 5.54723445002148534354769075602, 6.59377837076332882454431160866, 7.34724944707988836367594979061, 8.059690480081304334447558456761, 9.314800482843100546714400383107, 10.04246519271399080310404416660, 11.08537210879762789602904386651
