| L(s) = 1 | + 0.381·3-s + 1.38·5-s − 2.85·9-s − 3.23·13-s + 0.527·15-s + 17-s − 2.47·19-s + 5.23·23-s − 3.09·25-s − 2.23·27-s + 2.76·29-s + 9.56·31-s + 3.70·37-s − 1.23·39-s − 1.14·41-s + 4.85·43-s − 3.94·45-s + 10.9·47-s + 0.381·51-s + 12.3·53-s − 0.944·57-s + 5.23·59-s + 12.5·61-s − 4.47·65-s − 5.09·67-s + 2·69-s − 4.76·71-s + ⋯ |
| L(s) = 1 | + 0.220·3-s + 0.618·5-s − 0.951·9-s − 0.897·13-s + 0.136·15-s + 0.242·17-s − 0.567·19-s + 1.09·23-s − 0.618·25-s − 0.430·27-s + 0.513·29-s + 1.71·31-s + 0.609·37-s − 0.197·39-s − 0.178·41-s + 0.740·43-s − 0.587·45-s + 1.59·47-s + 0.0534·51-s + 1.69·53-s − 0.125·57-s + 0.681·59-s + 1.60·61-s − 0.554·65-s − 0.621·67-s + 0.240·69-s − 0.565·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.987570990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.987570990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + 4.76T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.724832633151008969326041321433, −7.939841840103145987156685408306, −7.15923958658699610569064113328, −6.29904525058262122446691698769, −5.63277144071430641358938984762, −4.88997697047779949498382368269, −3.94229319795948761347162377097, −2.73815711541078551966990732591, −2.33897299870422614144443771330, −0.820921076811590809590745864719,
0.820921076811590809590745864719, 2.33897299870422614144443771330, 2.73815711541078551966990732591, 3.94229319795948761347162377097, 4.88997697047779949498382368269, 5.63277144071430641358938984762, 6.29904525058262122446691698769, 7.15923958658699610569064113328, 7.939841840103145987156685408306, 8.724832633151008969326041321433
