| L(s) = 1 | − 1.71·2-s + 0.933·4-s − 3.13·7-s + 1.82·8-s + 2.19·11-s − 4.96·13-s + 5.37·14-s − 4.99·16-s + 1.99·17-s + 5.62·19-s − 3.75·22-s − 6.50·23-s + 8.50·26-s − 2.93·28-s − 2.30·29-s + 2.04·31-s + 4.90·32-s − 3.41·34-s + 6.99·37-s − 9.63·38-s − 10.4·41-s − 5.28·43-s + 2.04·44-s + 11.1·46-s + 9.28·47-s + 2.85·49-s − 4.63·52-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.466·4-s − 1.18·7-s + 0.645·8-s + 0.661·11-s − 1.37·13-s + 1.43·14-s − 1.24·16-s + 0.483·17-s + 1.29·19-s − 0.800·22-s − 1.35·23-s + 1.66·26-s − 0.553·28-s − 0.428·29-s + 0.366·31-s + 0.866·32-s − 0.585·34-s + 1.15·37-s − 1.56·38-s − 1.63·41-s − 0.806·43-s + 0.308·44-s + 1.64·46-s + 1.35·47-s + 0.407·49-s − 0.642·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5252920554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5252920554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 6.99T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 9.28T + 47T^{2} \) |
| 53 | \( 1 + 5.22T + 53T^{2} \) |
| 59 | \( 1 - 0.139T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 0.617T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 + 3.00T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 0.342T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.163370807446802233298166520138, −7.47846581518904719037858095489, −7.01050808241299688683111734603, −6.20913880858319197452788334836, −5.35184436779580647680467980313, −4.43064209372841727024129340851, −3.55159432068957399526721233063, −2.65645604381848202904964357184, −1.60633462479703528198692646371, −0.47055058099904807588053977423,
0.47055058099904807588053977423, 1.60633462479703528198692646371, 2.65645604381848202904964357184, 3.55159432068957399526721233063, 4.43064209372841727024129340851, 5.35184436779580647680467980313, 6.20913880858319197452788334836, 7.01050808241299688683111734603, 7.47846581518904719037858095489, 8.163370807446802233298166520138
