| 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
−11.08537210879762789602904386651, −10.04246519271399080310404416660, −9.314800482843100546714400383107, −8.059690480081304334447558456761, −7.34724944707988836367594979061, −6.59377837076332882454431160866, −5.54723445002148534354769075602, −4.83968165958510672625175717139, −2.84392553395533982714388459269, −1.70077118952805268797073047311,
1.09545611951571850860051351888, 2.48078367332796625636985215544, 3.63394855120813328861196177507, 5.45463642286968568228940211995, 5.78784221389158927500414696502, 6.80212394590505220957328642597, 8.143384692745678796053342149832, 9.269414677493465975754394303475, 10.02429911088573735654342960322, 10.75884519703677204930451968413
