L(s) = 1 | + 3.62i·5-s + i·7-s + 5.03·11-s − 3.12·13-s − 6.45i·17-s − 4i·19-s − 5.03·23-s − 8.12·25-s − 8.65i·29-s − 10.2i·31-s − 3.62·35-s + 2.87·37-s − 9.27i·41-s − 4i·43-s − 1.24·47-s + ⋯ |
L(s) = 1 | + 1.62i·5-s + 0.377i·7-s + 1.51·11-s − 0.866·13-s − 1.56i·17-s − 0.917i·19-s − 1.05·23-s − 1.62·25-s − 1.60i·29-s − 1.84i·31-s − 0.612·35-s + 0.472·37-s − 1.44i·41-s − 0.609i·43-s − 0.180·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.330701474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.330701474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 3.62iT - 5T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 6.45iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 8.65iT - 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 + 9.27iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 + 8.65iT - 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.12iT - 67T^{2} \) |
| 71 | \( 1 + 9.45T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 3.62iT - 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−8.237358469937702195737339480322, −7.28834294408793278465902773490, −7.00438341300407175065272320408, −6.23749879708412753043496319194, −5.54464832381696873443122967712, −4.34796397181857355512200672728, −3.71735835973257066784626858340, −2.56119245504847395030351656637, −2.27197125730796664090847950575, −0.38404535963594481566174694008,
1.34143856083689557305088106009, 1.56560843308815322639160300195, 3.31656711205520781316859669500, 4.18954117205344085272839837352, 4.60989716173377888521820748126, 5.58266454252805158801708670959, 6.25867338733977165491482547700, 7.10218448485266986678712072218, 8.055525114145052169532390517674, 8.558692924784204154769023046567