L(s) = 1 | + (−0.352 + 0.611i)3-s + (−0.5 − 0.866i)5-s + (0.647 − 2.56i)7-s + (1.25 + 2.16i)9-s + (2.25 − 3.89i)11-s + 5.09·13-s + 0.705·15-s + (−1 + 1.73i)17-s + (1.54 + 2.67i)19-s + (1.33 + 1.30i)21-s + (−2.89 − 5.01i)23-s + (−0.499 + 0.866i)25-s − 3.88·27-s + 9.50·29-s + (−2.70 + 4.68i)31-s + ⋯ |
L(s) = 1 | + (−0.203 + 0.352i)3-s + (−0.223 − 0.387i)5-s + (0.244 − 0.969i)7-s + (0.416 + 0.722i)9-s + (0.678 − 1.17i)11-s + 1.41·13-s + 0.182·15-s + (−0.242 + 0.420i)17-s + (0.354 + 0.614i)19-s + (0.292 + 0.283i)21-s + (−0.604 − 1.04i)23-s + (−0.0999 + 0.173i)25-s − 0.747·27-s + 1.76·29-s + (−0.485 + 0.841i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26666 - 0.198230i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26666 - 0.198230i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.647 + 2.56i)T \) |
good | 3 | \( 1 + (0.352 - 0.611i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.25 + 3.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 2.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.89 + 5.01i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9.50T + 29T^{2} \) |
| 31 | \( 1 + (2.70 - 4.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.54 + 6.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + (5.04 + 8.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.95 - 8.58i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.45 - 7.71i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0585 + 0.101i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (4.09 - 7.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.09 - 12.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + (2.04 + 3.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.63274242187820743390002973300, −10.74445708067111689060542361036, −10.27329657001604549897501625188, −8.718289668223804175196991506986, −8.184455428919357730852454491593, −6.84508144592908127245152150641, −5.75181646687244594181674559072, −4.41664561998213504208498772708, −3.62295099703294338690351473884, −1.26543248979319791345994143738,
1.65957801706493661487406782341, 3.37664838505248924435401190206, 4.71358651196764302376313780721, 6.16521403520588153953849409205, 6.81705453049961231551908910811, 8.011370545677994847968891202965, 9.131758329171966546045165460147, 9.874853288208778746271831505979, 11.31849868875329482345731270712, 11.82270515582679783150880397177