L(s) = 1 | + (−0.144 + 0.251i)2-s + (0.5 + 0.866i)3-s + (0.957 + 1.65i)4-s + (−1.16 + 2.01i)5-s − 0.289·6-s + (2.35 + 1.20i)7-s − 1.13·8-s + (−0.499 + 0.866i)9-s + (−0.337 − 0.584i)10-s + (−0.365 − 0.633i)11-s + (−0.957 + 1.65i)12-s − 0.115·13-s + (−0.644 + 0.415i)14-s − 2.32·15-s + (−1.75 + 3.03i)16-s + (−0.394 − 0.683i)17-s + ⋯ |
L(s) = 1 | + (−0.102 + 0.177i)2-s + (0.288 + 0.499i)3-s + (0.478 + 0.829i)4-s + (−0.520 + 0.900i)5-s − 0.118·6-s + (0.889 + 0.456i)7-s − 0.401·8-s + (−0.166 + 0.288i)9-s + (−0.106 − 0.184i)10-s + (−0.110 − 0.190i)11-s + (−0.276 + 0.478i)12-s − 0.0319·13-s + (−0.172 + 0.111i)14-s − 0.600·15-s + (−0.437 + 0.758i)16-s + (−0.0956 − 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.692433 + 1.37204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692433 + 1.37204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.35 - 1.20i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.144 - 0.251i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.16 - 2.01i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.365 + 0.633i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.115T + 13T^{2} \) |
| 17 | \( 1 + (0.394 + 0.683i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.46 + 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 + (0.747 + 1.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.07 - 3.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 + (-1.89 + 3.28i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.40 - 5.90i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.09 - 5.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 2.07i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.18 - 7.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + (4.95 + 8.57i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.241 - 0.419i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.84T + 83T^{2} \) |
| 89 | \( 1 + (1.53 - 2.65i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−11.31160177408830454436747899327, −10.69910204275552460906669649549, −9.292764041250214918869184227650, −8.559638852601304179574347096267, −7.56075724814783615962715982656, −7.09167059316652713928892535582, −5.69064736650993407166536496570, −4.42604076784451433152400194981, −3.27034053515498270791376095632, −2.43248964052498885208705690508,
0.988473831032917368374634401854, 2.02349503920634885577049846436, 3.79911245489724822720200784236, 5.02118598139517760877859211724, 5.88798133389985753014895233612, 7.24802133616272752463079227407, 7.87425752504670799702557893839, 8.839548897725342786777243323834, 9.787041490894708379543974108716, 10.77233910669579233470450501461