L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 − 1.73i)7-s + 0.999·8-s + (0.5 + 0.866i)11-s − 4·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s − 0.999·22-s + (−1.5 + 2.59i)23-s + (2.5 + 4.33i)25-s + (2 − 3.46i)26-s + (−0.5 + 2.59i)28-s + 9·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.150 + 0.261i)11-s − 1.10·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s − 0.213·22-s + (−0.312 + 0.541i)23-s + (0.5 + 0.866i)25-s + (0.392 − 0.679i)26-s + (−0.0944 + 0.490i)28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.100354199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100354199\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + 1.73i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.585008766208023078537445409950, −9.032450223150904502466295136007, −7.81468494840802803262887427405, −7.36183957453432619304249379309, −6.53821597116820243167318879751, −5.71654506427053333093812376522, −4.68973275864814302578313536073, −3.78214763964455677313804430139, −2.52303483453212598594153913163, −0.901863701225170832255194409845,
0.70268164688304263305963699137, 2.44406063584878451689783852979, 2.96060924653939199280351827165, 4.24670786852414636716958178473, 5.17090620876012841715518721636, 6.24288980124215225267153106198, 7.03823068505168467452240858567, 8.068247593891087323826028353156, 8.762017923723495760083096159468, 9.652948014956762117777238651995