L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 2.59i)7-s + 0.999·8-s + (1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (3.5 − 6.06i)17-s + (1.5 − 2.59i)18-s − 1.99·22-s + (−1.5 − 2.59i)23-s + (2 + 1.73i)28-s + 6·29-s + (3.5 − 6.06i)31-s + (−0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.188 − 0.981i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.848 − 1.47i)17-s + (0.353 − 0.612i)18-s − 0.426·22-s + (−0.312 − 0.541i)23-s + (0.377 + 0.327i)28-s + 1.11·29-s + (0.628 − 1.08i)31-s + (−0.0883 + 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.917926 - 0.698317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.917926 - 0.698317i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−11.29685536882335947917166577368, −10.24064832687969335178722925582, −9.829390911637106665841894047534, −8.441731629253052977476993093387, −7.68511200006167081488430338546, −6.72731984865423189383662011201, −5.08733248656995524803886834053, −4.11661633352941495922783940720, −2.75151428443267492863522121346, −1.05374713058931226300627199620,
1.62687141114993845125987658539, 3.55266354013516752286358624908, 4.92439453647826592655528096921, 6.04801085419458992481377128659, 6.80375593339708622047049576720, 8.047525054642078981280257566202, 8.786507306431124316737188132667, 9.742630520186105559131812883250, 10.45231566619500019162499392638, 11.97622288247141628084182420243