L(s) = 1 | + (−2.19 − 0.406i)5-s + (1.26 + 1.26i)7-s + 3.93i·11-s + (1.46 − 1.46i)13-s + (0.467 − 0.467i)17-s + 4.88i·19-s + (−0.707 − 0.707i)23-s + (4.66 + 1.78i)25-s + 0.536·29-s + 3.91·31-s + (−2.26 − 3.29i)35-s + (−0.291 − 0.291i)37-s − 11.7i·41-s + (−2.67 + 2.67i)43-s + (−7.55 + 7.55i)47-s + ⋯ |
L(s) = 1 | + (−0.983 − 0.181i)5-s + (0.477 + 0.477i)7-s + 1.18i·11-s + (0.407 − 0.407i)13-s + (0.113 − 0.113i)17-s + 1.12i·19-s + (−0.147 − 0.147i)23-s + (0.933 + 0.357i)25-s + 0.0996·29-s + 0.703·31-s + (−0.382 − 0.556i)35-s + (−0.0480 − 0.0480i)37-s − 1.83i·41-s + (−0.407 + 0.407i)43-s + (−1.10 + 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.051705101\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051705101\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.19 + 0.406i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1.26 - 1.26i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.93iT - 11T^{2} \) |
| 13 | \( 1 + (-1.46 + 1.46i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.467 + 0.467i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.88iT - 19T^{2} \) |
| 29 | \( 1 - 0.536T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 + (0.291 + 0.291i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.7iT - 41T^{2} \) |
| 43 | \( 1 + (2.67 - 2.67i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.55 - 7.55i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.0492 - 0.0492i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + (-0.268 - 0.268i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (4.49 - 4.49i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.65iT - 79T^{2} \) |
| 83 | \( 1 + (5.31 + 5.31i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \) |
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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
−8.466542999276704597692103657628, −7.987773428929244516814547150718, −7.36142513988197515568967332429, −6.54977110037005214860781998698, −5.59744832294826417723982378903, −4.85934022966481484741863787781, −4.15324159547039315314218088246, −3.36291378600948925193854001161, −2.26387806662818213265595218358, −1.22580658665447680944414401399,
0.33318504348644845139722014509, 1.41083678716499897610947652628, 2.87958535681050531383262861141, 3.49865247889890251622245240608, 4.40732649807812012891285541703, 4.97130120140988423832923726833, 6.13304518563216262099733165106, 6.71684119818836881986035509322, 7.54480773198460467438648397443, 8.206743700727112996889771584691