L(s) = 1 | + (0.309 − 0.951i)4-s + (−0.492 − 0.159i)7-s + (1.13 + 1.56i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.304 + 0.418i)28-s + (1.40 − 1.01i)31-s + 1.41i·43-s + (−0.592 − 0.430i)49-s + (1.83 − 0.596i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + 1.73·67-s + (−1.83 − 0.596i)73-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)4-s + (−0.492 − 0.159i)7-s + (1.13 + 1.56i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.304 + 0.418i)28-s + (1.40 − 1.01i)31-s + 1.41i·43-s + (−0.592 − 0.430i)49-s + (1.83 − 0.596i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + 1.73·67-s + (−1.83 − 0.596i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.372124585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.372124585\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.492 + 0.159i)T + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 1.56i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.40 + 1.01i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.83 + 0.596i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
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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
−8.870609657124705274573867854668, −7.988096721370573416170426353331, −6.99394491256407438496958343241, −6.40739113875133326004097908022, −5.94239978384049176612817710447, −4.86030043943429285393228253724, −4.16629354733893665600468378208, −3.07971541992418771274754003250, −2.01720155902631622372864514192, −0.983078397169420318390823652212,
1.25945269566180054280318475513, 2.77689284013895541461487162776, 3.28850976420990516536381320702, 3.98833281026919273058005822233, 5.29324493533744418782176899167, 5.86752371143551787277716215240, 6.82625993105732086858842380189, 7.45824956389277756248743175799, 8.273103022521806797360665315062, 8.639985437574688295331768699888