L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.363i)6-s − 1.61·7-s + (−0.809 − 0.587i)8-s + (−0.190 + 0.587i)9-s + (0.190 − 0.587i)12-s + (−0.500 − 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618·18-s + (0.809 − 0.587i)21-s + (−0.5 − 1.53i)23-s + 0.618·24-s + (−0.309 − 0.951i)27-s + (1.30 − 0.951i)28-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.363i)6-s − 1.61·7-s + (−0.809 − 0.587i)8-s + (−0.190 + 0.587i)9-s + (0.190 − 0.587i)12-s + (−0.500 − 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618·18-s + (0.809 − 0.587i)21-s + (−0.5 − 1.53i)23-s + 0.618·24-s + (−0.309 − 0.951i)27-s + (1.30 − 0.951i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1547038809\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1547038809\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−8.992524225984388873365467041634, −8.129029087979722495918571898673, −7.32791853162809149416096488965, −6.42759653189872476815767693084, −6.08077878203059521257669762957, −5.18435912036940852342520134532, −4.37639939261935441017678161702, −3.51816609724517987376546169782, −2.57085981066576402223042236850, −0.10226218936003877317716632035,
1.31329506776727121976079557250, 2.69945112830998560793613442524, 3.45798275297208945867002549464, 4.13660115052624248464680896495, 5.49351499201191168127081265547, 5.95594482080390037506365610999, 6.67109856615885111442917698745, 7.59034707351027836765961801597, 8.816091107816031871848806682945, 9.465381084049138238837142863626