L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.400 + 1.75i)5-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (1.62 − 0.781i)10-s + (−0.277 + 0.347i)13-s + (0.623 − 0.781i)16-s + 1.24·17-s + (0.623 − 0.781i)18-s + (−1.12 − 1.40i)20-s + (−2.02 + 0.974i)25-s + (0.400 + 0.193i)26-s + (−0.900 − 0.433i)32-s + (−0.277 − 1.21i)34-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.400 + 1.75i)5-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (1.62 − 0.781i)10-s + (−0.277 + 0.347i)13-s + (0.623 − 0.781i)16-s + 1.24·17-s + (0.623 − 0.781i)18-s + (−1.12 − 1.40i)20-s + (−2.02 + 0.974i)25-s + (0.400 + 0.193i)26-s + (−0.900 − 0.433i)32-s + (−0.277 − 1.21i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.121203848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.121203848\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 23 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 37 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
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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
−9.175403364784237571179910303186, −8.050255486303782597066166220230, −7.48672099894716418872958178484, −6.85214953102018346238944407294, −5.83018702091182914527600056238, −4.97991389072583732029196506121, −3.92946201051589895232279755522, −3.17911458557812779090465778757, −2.41156502618669270674940815144, −1.61333100877314964364469782377,
0.791797593988843908331206252063, 1.59795866858031388158649661879, 3.46053538549831930717725138565, 4.40068654994892696309010876951, 5.00214422105887108544579962481, 5.70706464801027237326288501363, 6.31715819848333832374393473624, 7.41746935130896715609711495067, 7.897850736487838868975780620394, 8.877769435419334062050107220299