L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.593·5-s + 0.999·8-s + (−0.296 + 0.514i)10-s + 0.593·11-s + (1.25 − 2.17i)13-s + (−0.5 + 0.866i)16-s + (1.46 − 2.52i)17-s + (−2.69 − 4.66i)19-s + (−0.296 − 0.514i)20-s + (−0.296 + 0.514i)22-s − 4.46·23-s − 4.64·25-s + (1.25 + 2.17i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.265·5-s + 0.353·8-s + (−0.0938 + 0.162i)10-s + 0.178·11-s + (0.348 − 0.603i)13-s + (−0.125 + 0.216i)16-s + (0.354 − 0.613i)17-s + (−0.617 − 1.06i)19-s + (−0.0663 − 0.114i)20-s + (−0.0632 + 0.109i)22-s − 0.930·23-s − 0.929·25-s + (0.246 + 0.427i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6884494532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6884494532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.593T + 5T^{2} \) |
| 11 | \( 1 - 0.593T + 11T^{2} \) |
| 13 | \( 1 + (-1.25 + 2.17i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.46 + 2.52i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.02 - 6.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.32 + 7.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 - 5.75i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (3.95 - 6.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 - 6.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.21 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.563240843114823842717470654543, −7.911577450235593896980349794058, −7.16095988197719403478849570789, −6.35572699595174127366338540263, −5.69573612586818489075352842846, −4.88553887340049754263921461611, −3.94711289109007552228311363075, −2.81836439384658533318556635292, −1.63946970784245799611940640114, −0.24790360963850521169569613457,
1.48248403363810641020347091196, 2.14046664033809257741214564003, 3.49369834366745149090777647308, 4.02935023614271208106957541005, 5.09896050266817734104276647099, 6.13387846949106982810454875505, 6.65704188780079613945160641449, 8.004723755064993841700854457100, 8.177368110730402319771363267458, 9.211343969721315373769634300418