L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.292 + 0.507i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 5.41·13-s − 0.585·15-s + (−3.12 − 5.40i)17-s + (1.41 − 2.44i)19-s + (1.82 − 3.16i)23-s + (2.32 + 4.03i)25-s − 0.999·27-s − 1.17·29-s + (−3.41 − 5.91i)31-s + (0.999 − 1.73i)33-s + (2 − 3.46i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.130 + 0.226i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 1.50·13-s − 0.151·15-s + (−0.757 − 1.31i)17-s + (0.324 − 0.561i)19-s + (0.381 − 0.660i)23-s + (0.465 + 0.806i)25-s − 0.192·27-s − 0.217·29-s + (−0.613 − 1.06i)31-s + (0.174 − 0.301i)33-s + (0.328 − 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.882532673\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.882532673\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.292 - 0.507i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + (3.12 + 5.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.82 + 3.16i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.41 + 5.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.87 - 3.25i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + (-2.94 - 5.10i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.17 + 2.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-2.87 + 4.98i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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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
−9.069015916072700014569291689525, −8.302503271208564284487486545145, −7.43631226993604205221436051271, −6.66276971195713430996892567061, −5.74059126780307498440962963580, −4.95532838166331572820802489256, −3.99076366076338142621916970024, −3.21405128954750419942968365275, −2.32876643329338936231765442298, −0.70887067804965843296346965018,
1.16205599674845513647561804054, 2.05118661902886876074088602030, 3.33196213394815158410855379853, 4.03890435393439115046861694035, 5.09463882897299474946038088414, 6.08832715504166373699955198292, 6.62915027159913340096840531387, 7.61684821961654307893374423423, 8.292153954544744291944235580449, 8.827090754214502111043459927860