L(s) = 1 | + (−1.60 − 2.20i)2-s + (−0.951 + 0.309i)3-s + (−1.67 + 5.16i)4-s + (−2.23 − 0.0757i)5-s + (2.20 + 1.60i)6-s + (0.462 + 0.150i)7-s + (8.89 − 2.88i)8-s + (0.809 − 0.587i)9-s + (3.41 + 5.04i)10-s + (3.06 + 1.26i)11-s − 5.43i·12-s + (2.12 + 2.92i)13-s + (−0.409 − 1.26i)14-s + (2.14 − 0.618i)15-s + (−11.8 − 8.59i)16-s + (−2.14 + 2.94i)17-s + ⋯ |
L(s) = 1 | + (−1.13 − 1.55i)2-s + (−0.549 + 0.178i)3-s + (−0.839 + 2.58i)4-s + (−0.999 − 0.0338i)5-s + (0.900 + 0.654i)6-s + (0.174 + 0.0568i)7-s + (3.14 − 1.02i)8-s + (0.269 − 0.195i)9-s + (1.07 + 1.59i)10-s + (0.924 + 0.381i)11-s − 1.56i·12-s + (0.588 + 0.810i)13-s + (−0.109 − 0.336i)14-s + (0.554 − 0.159i)15-s + (−2.95 − 2.14i)16-s + (−0.519 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.404546 - 0.0273474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.404546 - 0.0273474i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (2.23 + 0.0757i)T \) |
| 11 | \( 1 + (-3.06 - 1.26i)T \) |
good | 2 | \( 1 + (1.60 + 2.20i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.462 - 0.150i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 2.92i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.14 - 2.94i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.504 - 1.55i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 3.54iT - 23T^{2} \) |
| 29 | \( 1 + (0.326 - 1.00i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-8.17 - 2.65i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.39 - 10.4i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.41iT - 43T^{2} \) |
| 47 | \( 1 + (4.95 - 1.61i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.875 + 1.20i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.46 + 7.58i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.78 - 6.38i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 0.432iT - 67T^{2} \) |
| 71 | \( 1 + (4.86 + 3.53i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.10 - 0.359i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (3.57 - 2.60i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.86 - 2.57i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + (-0.822 - 1.13i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.29487997685276252954406793904, −11.40415530514166219765935981873, −11.17644766649630831648667742558, −9.895897112441247042692360248133, −8.984661538839551309480688126325, −8.072398721793404315939753918594, −6.84723302400776284242163854752, −4.40536884678750651847589388498, −3.58013885831474209693525103940, −1.49976851255489888021833872316,
0.67260973755935056253684694863, 4.40721581348214180297179848456, 5.73392295349008015950369209737, 6.73833954258528005486956287558, 7.60928947789463730392070533224, 8.498331085559639484454120352128, 9.436235379879075018131457956781, 10.80134342770424330041671884649, 11.41995535178042422596901758184, 13.03129617314872819539789890228