| L(s) = 1 | − 0.222·3-s + 0.0345·5-s − 5.16·7-s − 2.95·9-s + 4.32·11-s − 1.13·13-s − 0.00766·15-s + 6.24·17-s + 0.549·19-s + 1.14·21-s − 1.07·23-s − 4.99·25-s + 1.32·27-s − 8.90·29-s − 7.44·31-s − 0.959·33-s − 0.178·35-s − 10.7·37-s + 0.251·39-s + 0.759·41-s + 4.66·43-s − 0.101·45-s + 9.48·47-s + 19.7·49-s − 1.38·51-s − 9.26·53-s + 0.149·55-s + ⋯ |
| L(s) = 1 | − 0.128·3-s + 0.0154·5-s − 1.95·7-s − 0.983·9-s + 1.30·11-s − 0.314·13-s − 0.00197·15-s + 1.51·17-s + 0.126·19-s + 0.250·21-s − 0.223·23-s − 0.999·25-s + 0.254·27-s − 1.65·29-s − 1.33·31-s − 0.166·33-s − 0.0301·35-s − 1.77·37-s + 0.0403·39-s + 0.118·41-s + 0.710·43-s − 0.0151·45-s + 1.38·47-s + 2.81·49-s − 0.194·51-s − 1.27·53-s + 0.0201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8490058316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8490058316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.222T + 3T^{2} \) |
| 5 | \( 1 - 0.0345T + 5T^{2} \) |
| 7 | \( 1 + 5.16T + 7T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 0.549T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 + 8.90T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 0.759T + 41T^{2} \) |
| 43 | \( 1 - 4.66T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 9.26T + 53T^{2} \) |
| 59 | \( 1 - 0.198T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 0.237T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 0.602T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−7.61812107658522946264694638892, −7.14294936082188032899925946471, −6.34306734422802028420193896480, −5.77870614859643794725793668541, −5.39566696438479835600184430802, −3.81913720988185867357580551752, −3.66001048237188500986737928706, −2.87244195304309567815009540322, −1.77398952756555992280197637771, −0.44304430041564088541315086715,
0.44304430041564088541315086715, 1.77398952756555992280197637771, 2.87244195304309567815009540322, 3.66001048237188500986737928706, 3.81913720988185867357580551752, 5.39566696438479835600184430802, 5.77870614859643794725793668541, 6.34306734422802028420193896480, 7.14294936082188032899925946471, 7.61812107658522946264694638892
