| L(s) = 1 | − 1.83·2-s − 1.24·3-s + 1.36·4-s + 5-s + 2.28·6-s − 7-s + 1.16·8-s − 1.45·9-s − 1.83·10-s − 1.69·12-s + 2.32·13-s + 1.83·14-s − 1.24·15-s − 4.86·16-s − 4.59·17-s + 2.66·18-s − 7.12·19-s + 1.36·20-s + 1.24·21-s − 9.05·23-s − 1.45·24-s + 25-s − 4.26·26-s + 5.53·27-s − 1.36·28-s + 4.99·29-s + 2.28·30-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.717·3-s + 0.681·4-s + 0.447·5-s + 0.930·6-s − 0.377·7-s + 0.413·8-s − 0.484·9-s − 0.579·10-s − 0.489·12-s + 0.644·13-s + 0.490·14-s − 0.321·15-s − 1.21·16-s − 1.11·17-s + 0.628·18-s − 1.63·19-s + 0.304·20-s + 0.271·21-s − 1.88·23-s − 0.296·24-s + 0.200·25-s − 0.836·26-s + 1.06·27-s − 0.257·28-s + 0.927·29-s + 0.416·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2799706202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2799706202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 9.05T + 23T^{2} \) |
| 29 | \( 1 - 4.99T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 7.35T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 + 0.0843T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.309472160780640948531410848583, −8.100306153362646557420406388094, −6.75048849454844534121402795528, −6.42880361265492780572762724304, −5.73418519676335572988152979555, −4.67287816133126691125613194829, −3.91533985718281024244598487122, −2.48351880064679004705550139549, −1.72977210037452811745789985480, −0.36948390590416860017314476564,
0.36948390590416860017314476564, 1.72977210037452811745789985480, 2.48351880064679004705550139549, 3.91533985718281024244598487122, 4.67287816133126691125613194829, 5.73418519676335572988152979555, 6.42880361265492780572762724304, 6.75048849454844534121402795528, 8.100306153362646557420406388094, 8.309472160780640948531410848583
