| L(s) = 1 | + 1.85·2-s − 2.29·3-s + 1.45·4-s + 0.197·5-s − 4.26·6-s − 1.01·8-s + 2.26·9-s + 0.366·10-s − 4.18·11-s − 3.33·12-s − 0.452·15-s − 4.79·16-s − 0.841·17-s + 4.20·18-s − 1.35·19-s + 0.286·20-s − 7.77·22-s − 4.11·23-s + 2.33·24-s − 4.96·25-s + 1.69·27-s − 8.23·29-s − 0.841·30-s + 1.28·31-s − 6.87·32-s + 9.59·33-s − 1.56·34-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.32·3-s + 0.726·4-s + 0.0882·5-s − 1.74·6-s − 0.359·8-s + 0.754·9-s + 0.115·10-s − 1.26·11-s − 0.962·12-s − 0.116·15-s − 1.19·16-s − 0.204·17-s + 0.991·18-s − 0.310·19-s + 0.0641·20-s − 1.65·22-s − 0.858·23-s + 0.476·24-s − 0.992·25-s + 0.325·27-s − 1.52·29-s − 0.153·30-s + 0.230·31-s − 1.21·32-s + 1.67·33-s − 0.268·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039757426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039757426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 - 0.197T + 5T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 17 | \( 1 + 0.841T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 5.97T + 71T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.52665365365027129248681567945, −6.80453853000988585203960840215, −6.01331977192430044770319220189, −5.60266096533168113733757211110, −5.24648345381796783057927092312, −4.34617728012621255226079421238, −3.87397872679358873081960501599, −2.76490483130386600155772442940, −2.02368294629880731046909386675, −0.41161646736622487557107309234,
0.41161646736622487557107309234, 2.02368294629880731046909386675, 2.76490483130386600155772442940, 3.87397872679358873081960501599, 4.34617728012621255226079421238, 5.24648345381796783057927092312, 5.60266096533168113733757211110, 6.01331977192430044770319220189, 6.80453853000988585203960840215, 7.52665365365027129248681567945
