L(s) = 1 | + (−0.903 + 1.08i)2-s + (−0.707 + 0.707i)3-s + (−0.368 − 1.96i)4-s + (0.770 + 2.09i)5-s + (−0.130 − 1.40i)6-s − 3.05·7-s + (2.47 + 1.37i)8-s − 1.00i·9-s + (−2.98 − 1.05i)10-s + (−1.80 + 1.80i)11-s + (1.65 + 1.12i)12-s + (−2.47 + 2.47i)13-s + (2.75 − 3.31i)14-s + (−2.02 − 0.939i)15-s + (−3.72 + 1.44i)16-s − 3.66i·17-s + ⋯ |
L(s) = 1 | + (−0.638 + 0.769i)2-s + (−0.408 + 0.408i)3-s + (−0.184 − 0.982i)4-s + (0.344 + 0.938i)5-s + (−0.0533 − 0.574i)6-s − 1.15·7-s + (0.873 + 0.486i)8-s − 0.333i·9-s + (−0.942 − 0.334i)10-s + (−0.545 + 0.545i)11-s + (0.476 + 0.326i)12-s + (−0.685 + 0.685i)13-s + (0.736 − 0.887i)14-s + (−0.523 − 0.242i)15-s + (−0.932 + 0.361i)16-s − 0.889i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0618446 - 0.372086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0618446 - 0.372086i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.903 - 1.08i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.770 - 2.09i)T \) |
good | 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 + (1.80 - 1.80i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.47 - 2.47i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.66iT - 17T^{2} \) |
| 19 | \( 1 + (2.31 + 2.31i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + (-4.74 - 4.74i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + (5.40 + 5.40i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.47iT - 41T^{2} \) |
| 43 | \( 1 + (-4.19 - 4.19i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (-9.99 - 9.99i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.47 - 2.47i)T - 59iT^{2} \) |
| 61 | \( 1 + (-8.01 - 8.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.60 + 8.60i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.63iT - 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + (3.65 - 3.65i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67449835199358099026308624903, −11.40802035174568322953237623205, −10.31371015750636673665011732741, −9.869153174112453999014236353133, −9.013473358301694092262647703086, −7.38165511029501237966327424122, −6.73856302289206255827347387141, −5.83421753530612101429801524937, −4.52661577897402026358391787011, −2.61921657530257452244795271501,
0.36909116130108475020660233397, 2.24640102974485943662964059021, 3.81866774609572367839427672199, 5.39934932365196591991173114469, 6.55315685310259035718381163336, 8.000718808825757913298220966467, 8.646746280145142773774007522736, 10.10436032560283280826445255791, 10.20912019782762623489316261807, 11.80946836572002662956663438219