L(s) = 1 | + (1.33 + 0.473i)2-s + (1.55 + 1.26i)4-s + (−1.45 + 1.69i)5-s + (1.53 + 1.53i)7-s + (1.47 + 2.41i)8-s + (−2.74 + 1.57i)10-s − 2.72·11-s + (−0.857 − 0.857i)13-s + (1.31 + 2.76i)14-s + (0.818 + 3.91i)16-s + (2.55 − 2.55i)17-s + 3.54·19-s + (−4.39 + 0.803i)20-s + (−3.63 − 1.28i)22-s + (−0.626 − 0.626i)23-s + ⋯ |
L(s) = 1 | + (0.942 + 0.334i)2-s + (0.776 + 0.630i)4-s + (−0.650 + 0.759i)5-s + (0.578 + 0.578i)7-s + (0.520 + 0.853i)8-s + (−0.866 + 0.498i)10-s − 0.821·11-s + (−0.237 − 0.237i)13-s + (0.351 + 0.738i)14-s + (0.204 + 0.978i)16-s + (0.619 − 0.619i)17-s + 0.812·19-s + (−0.983 + 0.179i)20-s + (−0.774 − 0.274i)22-s + (−0.130 − 0.130i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69015 + 1.35056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69015 + 1.35056i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.33 - 0.473i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.45 - 1.69i)T \) |
good | 7 | \( 1 + (-1.53 - 1.53i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + (0.857 + 0.857i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.55 + 2.55i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + (0.626 + 0.626i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.12iT - 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 + (-4.21 + 4.21i)T - 37iT^{2} \) |
| 41 | \( 1 + 12.4iT - 41T^{2} \) |
| 43 | \( 1 + (5.67 + 5.67i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.45 - 9.45i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.46 + 6.46i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.51iT - 59T^{2} \) |
| 61 | \( 1 + 9.49iT - 61T^{2} \) |
| 67 | \( 1 + (9.91 - 9.91i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.19iT - 71T^{2} \) |
| 73 | \( 1 + (5.71 - 5.71i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.7iT - 79T^{2} \) |
| 83 | \( 1 + (-3.58 + 3.58i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + (1.29 + 1.29i)T + 97iT^{2} \) |
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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
−11.80122543221977983531328725865, −11.01667079090759332609803516964, −10.06971103477239995351481469515, −8.443799435696119000925469686023, −7.68622479038374122712188548059, −6.91586844031587533113790424905, −5.62042840219375674439838608257, −4.84099197842392325085850169831, −3.43810550037220053799946581816, −2.48130970836442158315265856456,
1.28392326376367360080991761305, 3.07165599833003541363459053894, 4.36523048994677469062231192784, 4.97506450710618713276985219840, 6.18020953022194006287758477580, 7.58878875913903887773805914778, 8.109909182027990948742258559773, 9.711401994364331970671458678969, 10.48309108542729288343873508624, 11.67943776211302712260924574375