| L(s) = 1 | + 1.30·2-s + 2.30·3-s − 0.302·4-s − 1.30·5-s + 3·6-s + 3.30·7-s − 3·8-s + 2.30·9-s − 1.69·10-s + 3·11-s − 0.697·12-s + 0.302·13-s + 4.30·14-s − 3·15-s − 3.30·16-s + 3.00·18-s − 4.90·19-s + 0.394·20-s + 7.60·21-s + 3.90·22-s − 1.30·23-s − 6.90·24-s − 3.30·25-s + 0.394·26-s − 1.60·27-s − 1.00·28-s − 0.908·29-s + ⋯ |
| L(s) = 1 | + 0.921·2-s + 1.32·3-s − 0.151·4-s − 0.582·5-s + 1.22·6-s + 1.24·7-s − 1.06·8-s + 0.767·9-s − 0.536·10-s + 0.904·11-s − 0.201·12-s + 0.0839·13-s + 1.14·14-s − 0.774·15-s − 0.825·16-s + 0.707·18-s − 1.12·19-s + 0.0882·20-s + 1.65·21-s + 0.833·22-s − 0.271·23-s − 1.41·24-s − 0.660·25-s + 0.0773·26-s − 0.308·27-s − 0.188·28-s − 0.168·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.554270813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554270813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 0.302T + 13T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 6.60T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 5.39T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.60T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 0.0916T + 97T^{2} \) |
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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
−11.94374757056220458632310727812, −11.19950305010814087685810088064, −9.696377167794522553231594162279, −8.581101562054476809953256854122, −8.313389711128422853170027478833, −6.99904254745467847319296851001, −5.49183445926367326829993812509, −4.20727869685272104193940726578, −3.69232460168068043981243395288, −2.12906324888282169970633215355,
2.12906324888282169970633215355, 3.69232460168068043981243395288, 4.20727869685272104193940726578, 5.49183445926367326829993812509, 6.99904254745467847319296851001, 8.313389711128422853170027478833, 8.581101562054476809953256854122, 9.696377167794522553231594162279, 11.19950305010814087685810088064, 11.94374757056220458632310727812
