L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.292 + 0.707i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.292 − 0.707i)10-s + 2i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s − 1.00·18-s + (0.707 + 0.292i)20-s + (0.292 + 0.292i)25-s + (−1.41 − 1.41i)26-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.292 + 0.707i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.292 − 0.707i)10-s + 2i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s − 1.00·18-s + (0.707 + 0.292i)20-s + (0.292 + 0.292i)25-s + (−1.41 − 1.41i)26-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8135887948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8135887948\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
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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
−8.963496633033905233351998596253, −8.333428804253593515343779294271, −7.39305428322575258687015400428, −7.07382149818057030399372759746, −6.36371574000492539707924882567, −5.48542224060844217595885183287, −4.51232389073341059948598171692, −3.83606717947741136732340941936, −2.31257225325893339478800050740, −1.53677249224403310468810495226,
0.67462249582215132399735969443, 1.51765393225396190521489993603, 3.19974757538479817229075309760, 3.29677126391270769567097625217, 4.69926971905537516192390360629, 5.18765683861963310014032409969, 6.52283731195646206340894548442, 7.23086131065320965486865430661, 8.114202200424037202646084147307, 8.439268726153336562581636659625