| L(s) = 1 | + i·3-s − 3.09·5-s − 0.646i·7-s − 9-s − 1.41·11-s + (−2.44 + 2.64i)13-s − 3.09i·15-s + 3.58·17-s − 1.11·19-s + 0.646·21-s − 3.46·23-s + 4.58·25-s − i·27-s − 0.913i·29-s + 1.93i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.38·5-s − 0.244i·7-s − 0.333·9-s − 0.426·11-s + (−0.679 + 0.733i)13-s − 0.799i·15-s + 0.868·17-s − 0.256·19-s + 0.140·21-s − 0.722·23-s + 0.916·25-s − 0.192i·27-s − 0.169i·29-s + 0.348i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8124453480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8124453480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 13 | \( 1 + (2.44 - 2.64i)T \) |
| good | 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 + 0.646iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 0.913iT - 29T^{2} \) |
| 31 | \( 1 - 1.93iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5.95iT - 41T^{2} \) |
| 43 | \( 1 - 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 3.74iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 + 0.913iT - 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 2.44iT - 71T^{2} \) |
| 73 | \( 1 + 5.65iT - 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 8.78iT - 89T^{2} \) |
| 97 | \( 1 - 7.89iT - 97T^{2} \) |
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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
−8.713403627344371392037545500566, −8.129916324626559431050743982750, −7.40936832645983000607868994415, −6.75762566528963679208788470341, −5.53892622087758311934748506991, −4.78220574277251264710294234219, −3.95331998428500178216692188029, −3.43356475367658023592664301336, −2.17444374905177892794163335278, −0.39454456697497826768381038760,
0.77990225607006776698718052134, 2.31671296925202302830387841727, 3.24126331182192268712067456506, 4.08032937300687739376081332265, 5.10565382596651613845414351682, 5.83565923389472633068921758761, 6.90590662195041925025765174150, 7.62980860368579331090131873086, 8.007823456648393744546387402924, 8.687563488632843495292539802200
