L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.22 − 3.85i)7-s + 0.999·8-s + (0.724 + 1.25i)11-s + (1.22 − 2.12i)13-s + (−2.22 + 3.85i)14-s + (−0.5 − 0.866i)16-s − 3.89·17-s − 0.550·19-s + (0.724 − 1.25i)22-s + (−1.44 + 2.51i)23-s − 2.44·26-s + 4.44·28-s + (−3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.840 − 1.45i)7-s + 0.353·8-s + (0.218 + 0.378i)11-s + (0.339 − 0.588i)13-s + (−0.594 + 1.02i)14-s + (−0.125 − 0.216i)16-s − 0.945·17-s − 0.126·19-s + (0.154 − 0.267i)22-s + (−0.302 + 0.523i)23-s − 0.480·26-s + 0.840·28-s + (−0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2322735495\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2322735495\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.22 + 3.85i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.724 - 1.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 2.12i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.89T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 + (1.44 - 2.51i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.72 + 6.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.224 - 0.389i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 + (5.62 - 9.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.224 - 0.389i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.72 - 8.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.359494106142471835150129652541, −8.368855349179045820652475702607, −7.45677186386231722785052385803, −6.85934787301664589727153291607, −5.85713378690841019442814341864, −4.45064048363137946074938506038, −3.84117635950114504206095144400, −2.87188775840963021039364430964, −1.44811142962811307830475593414, −0.10709028405462893851947625571,
1.89435082893973505463307238468, 3.02165608148900063760570956212, 4.25877510313444359586355744192, 5.34662542325445546372725553863, 6.27508099964263242612481768377, 6.52014629898770229784946468461, 7.78467501091490859845868978461, 8.640157349153905076481817770887, 9.224338856328760738210376013067, 9.670696213922993804574429650325