L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.508 − 0.293i)3-s + (−0.499 + 0.866i)4-s + (−0.863 − 1.49i)5-s + 0.587i·6-s + (1.63 + 2.08i)7-s + 0.999·8-s + (−1.32 − 2.29i)9-s + (−0.863 + 1.49i)10-s + (−3.46 − 2.00i)11-s + (0.508 − 0.293i)12-s − 5.60i·13-s + (0.987 − 2.45i)14-s + 1.01i·15-s + (−0.5 − 0.866i)16-s + (−2.91 + 5.05i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.293 − 0.169i)3-s + (−0.249 + 0.433i)4-s + (−0.386 − 0.668i)5-s + 0.239i·6-s + (0.616 + 0.787i)7-s + 0.353·8-s + (−0.442 − 0.766i)9-s + (−0.273 + 0.472i)10-s + (−1.04 − 0.603i)11-s + (0.146 − 0.0848i)12-s − 1.55i·13-s + (0.263 − 0.656i)14-s + 0.262i·15-s + (−0.125 − 0.216i)16-s + (−0.708 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0572050 - 0.544492i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0572050 - 0.544492i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.63 - 2.08i)T \) |
| 23 | \( 1 + (4.78 + 0.280i)T \) |
good | 3 | \( 1 + (0.508 + 0.293i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.863 + 1.49i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.46 + 2.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.60iT - 13T^{2} \) |
| 17 | \( 1 + (2.91 - 5.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.863 + 1.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 9.71T + 29T^{2} \) |
| 31 | \( 1 + (-3.47 - 2.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.43 + 4.29i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.60iT - 41T^{2} \) |
| 43 | \( 1 + 3.11iT - 43T^{2} \) |
| 47 | \( 1 + (-6.35 + 3.66i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.09 + 1.20i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.45 - 3.14i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.19 + 2.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.25 + 5.34i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + (-3.47 - 2.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.75 + 5.05i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + (-7.80 - 13.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−11.10400709566088766153202903986, −10.56785288813805026951980282041, −9.169187987620279547931327403974, −8.372017000002904639016882549106, −7.85628428846234382286709490721, −6.01030835006959695687821673838, −5.24626659549282581338497283717, −3.78187373756057932638848036460, −2.37065379103877837370653107511, −0.43648854601517338674010776772,
2.23679816261749809671770551628, 4.24736960363053217849267307436, 5.03981715019455921972491579929, 6.42053302494098333986090778955, 7.47673351008816375211714828524, 7.87985256449055388886211994283, 9.292063521349645606223830781016, 10.23028182413659980707690192755, 11.18893754217880198612233364998, 11.53343044129534152058598713400