L(s) = 1 | + (0.618 − 1.90i)2-s + (0.809 − 0.587i)3-s + (−1.61 − 1.17i)4-s + (0.309 + 0.951i)5-s + (−0.618 − 1.90i)6-s + (−1.61 − 1.17i)7-s + (−0.618 + 1.90i)9-s + 1.99·10-s − 1.99·12-s + (−1.23 + 3.80i)13-s + (−3.23 + 2.35i)14-s + (0.809 + 0.587i)15-s + (−1.23 − 3.80i)16-s + (0.618 + 1.90i)17-s + (3.23 + 2.35i)18-s + ⋯ |
L(s) = 1 | + (0.437 − 1.34i)2-s + (0.467 − 0.339i)3-s + (−0.809 − 0.587i)4-s + (0.138 + 0.425i)5-s + (−0.252 − 0.776i)6-s + (−0.611 − 0.444i)7-s + (−0.206 + 0.634i)9-s + 0.632·10-s − 0.577·12-s + (−0.342 + 1.05i)13-s + (−0.864 + 0.628i)14-s + (0.208 + 0.151i)15-s + (−0.309 − 0.951i)16-s + (0.149 + 0.461i)17-s + (0.762 + 0.554i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.914628 - 1.09063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.914628 - 1.09063i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.618 + 1.90i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.309 - 0.951i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.61 + 1.17i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (1.23 - 3.80i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.618 - 1.90i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.16 + 6.65i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.42 + 1.76i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.47 - 4.70i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.85 - 5.70i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (4.04 + 2.93i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.70 + 11.4i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + (0.927 + 2.85i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.23 - 2.35i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.09 + 9.51i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.85 - 5.70i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (2.16 - 6.65i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.15775002120475738175601832186, −12.19819170035980524795334847302, −11.13694118401869825506872316173, −10.30764802693525205854483697217, −9.321866092154329620502773866796, −7.76199415890923757286587926662, −6.54419353677406404941339950950, −4.59935306323874333126664031857, −3.24192834834163559046372978032, −2.02866506823839859598279432937,
3.22592864038877381696018567953, 4.90425149882598123780545724592, 5.91649193120471835366207873474, 7.04946400316430978006336104836, 8.336856758388475125966117588826, 9.155632202663895206372875764647, 10.35201309558406311667888410672, 12.08526684871614625099246303759, 13.02181320360794293250751067355, 14.03201478379900984676172749211