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
−8.439268726153336562581636659625, −8.114202200424037202646084147307, −7.23086131065320965486865430661, −6.52283731195646206340894548442, −5.18765683861963310014032409969, −4.69926971905537516192390360629, −3.29677126391270769567097625217, −3.19974757538479817229075309760, −1.51765393225396190521489993603, −0.67462249582215132399735969443,
1.53677249224403310468810495226, 2.31257225325893339478800050740, 3.83606717947741136732340941936, 4.51232389073341059948598171692, 5.48542224060844217595885183287, 6.36371574000492539707924882567, 7.07382149818057030399372759746, 7.39305428322575258687015400428, 8.333428804253593515343779294271, 8.963496633033905233351998596253