| L(s) = 1 | − 1.99·3-s + 2.93·5-s − 3.10·7-s + 0.987·9-s − 3.97·11-s − 5.28·13-s − 5.86·15-s − 1.60·17-s − 5.99·19-s + 6.19·21-s + 0.369·23-s + 3.62·25-s + 4.01·27-s − 3.25·29-s − 2.38·31-s + 7.94·33-s − 9.11·35-s − 11.2·37-s + 10.5·39-s − 4.05·41-s − 2.21·43-s + 2.90·45-s + 1.67·47-s + 2.62·49-s + 3.19·51-s − 8.55·53-s − 11.6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.31·5-s − 1.17·7-s + 0.329·9-s − 1.19·11-s − 1.46·13-s − 1.51·15-s − 0.388·17-s − 1.37·19-s + 1.35·21-s + 0.0771·23-s + 0.725·25-s + 0.773·27-s − 0.603·29-s − 0.427·31-s + 1.38·33-s − 1.54·35-s − 1.84·37-s + 1.69·39-s − 0.633·41-s − 0.338·43-s + 0.432·45-s + 0.244·47-s + 0.375·49-s + 0.447·51-s − 1.17·53-s − 1.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1978200112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1978200112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 13 | \( 1 + 5.28T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 - 0.369T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 - 7.59T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 2.60T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−7.64183690260196635442063105596, −6.71205390039518137839312423488, −6.52747759470576544196921615096, −5.66361364028964407573259851097, −5.25271283720160166526779847726, −4.66401906335553748817622346267, −3.39137924557699273182652501820, −2.48862574525061154727582659135, −1.91763725215237820258556204599, −0.21404784451318173930920510098,
0.21404784451318173930920510098, 1.91763725215237820258556204599, 2.48862574525061154727582659135, 3.39137924557699273182652501820, 4.66401906335553748817622346267, 5.25271283720160166526779847726, 5.66361364028964407573259851097, 6.52747759470576544196921615096, 6.71205390039518137839312423488, 7.64183690260196635442063105596
