| L(s) = 1 | + 3-s + 0.585·5-s − 1.41·7-s + 9-s + 5.65·11-s + 5.65·13-s + 0.585·15-s − 2.24·17-s − 8.24·19-s − 1.41·21-s + 23-s − 4.65·25-s + 27-s + 8.82·29-s − 1.17·31-s + 5.65·33-s − 0.828·35-s − 3.17·37-s + 5.65·39-s − 2·41-s − 5.41·43-s + 0.585·45-s − 10.4·47-s − 5·49-s − 2.24·51-s + 4.58·53-s + 3.31·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.261·5-s − 0.534·7-s + 0.333·9-s + 1.70·11-s + 1.56·13-s + 0.151·15-s − 0.543·17-s − 1.89·19-s − 0.308·21-s + 0.208·23-s − 0.931·25-s + 0.192·27-s + 1.63·29-s − 0.210·31-s + 0.984·33-s − 0.140·35-s − 0.521·37-s + 0.905·39-s − 0.312·41-s − 0.825·43-s + 0.0873·45-s − 1.52·47-s − 0.714·49-s − 0.314·51-s + 0.629·53-s + 0.446·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620236250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620236250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−11.90050735963500801445277864692, −10.90679843816514506236089674569, −9.882319819910005822888162400279, −8.872006503893159585757153755981, −8.384701073411926562626182233033, −6.60296333474831789043564547276, −6.31670944000890475573566842943, −4.35367144990027765798827833257, −3.43589858415556779874635487995, −1.70534628834130459930649122147,
1.70534628834130459930649122147, 3.43589858415556779874635487995, 4.35367144990027765798827833257, 6.31670944000890475573566842943, 6.60296333474831789043564547276, 8.384701073411926562626182233033, 8.872006503893159585757153755981, 9.882319819910005822888162400279, 10.90679843816514506236089674569, 11.90050735963500801445277864692
