L(s) = 1 | + (−2 − 3.46i)5-s + (1 − 1.73i)7-s + (−2 + 3.46i)11-s + (−0.5 − 0.866i)13-s − 2·17-s − 2·19-s + (−5.49 + 9.52i)25-s + (−3 + 5.19i)29-s + (5 + 8.66i)31-s − 7.99·35-s + 10·37-s + (4 + 6.92i)41-s + (−2 + 3.46i)43-s + (−2 + 3.46i)47-s + (1.50 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.894 − 1.54i)5-s + (0.377 − 0.654i)7-s + (−0.603 + 1.04i)11-s + (−0.138 − 0.240i)13-s − 0.485·17-s − 0.458·19-s + (−1.09 + 1.90i)25-s + (−0.557 + 0.964i)29-s + (0.898 + 1.55i)31-s − 1.35·35-s + 1.64·37-s + (0.624 + 1.08i)41-s + (−0.304 + 0.528i)43-s + (−0.291 + 0.505i)47-s + (0.214 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.054791164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054791164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.365687869088255055242622477368, −7.74501738217377663438555489806, −7.29198842071995452148152637169, −6.24951028845379239868272080901, −5.09433469601001017285446215328, −4.65738658527342304637210119167, −4.22266355493436418796376542636, −3.08084263456936232562721215805, −1.76279008618299495525489044732, −0.849855161597366147245000030104,
0.39078062672676535211292133237, 2.44110383121023674656606533326, 2.60583150622207414029074872257, 3.82424775745718412502472350499, 4.32514203217623369305126411565, 5.70085547701125098256832548416, 6.06854104351493415325491798050, 7.01981138699950072207736125587, 7.62099377597551170679067110802, 8.259923108125085780991165032765