| L(s) = 1 | + 0.765·3-s + 1.84·5-s + 0.317·7-s − 2.41·9-s + 0.765·11-s + 1.41·15-s − 2.24·19-s + 0.242·21-s + 6.62·23-s − 1.58·25-s − 4.14·27-s + 5.54·29-s + 2.29·31-s + 0.585·33-s + 0.585·35-s + 0.765·37-s − 2.48·41-s + 10.2·43-s − 4.46·45-s + 12.8·47-s − 6.89·49-s − 3.07·53-s + 1.41·55-s − 1.71·57-s − 7.89·59-s + 9.87·61-s − 0.765·63-s + ⋯ |
| L(s) = 1 | + 0.441·3-s + 0.826·5-s + 0.119·7-s − 0.804·9-s + 0.230·11-s + 0.365·15-s − 0.514·19-s + 0.0529·21-s + 1.38·23-s − 0.317·25-s − 0.797·27-s + 1.02·29-s + 0.412·31-s + 0.101·33-s + 0.0990·35-s + 0.125·37-s − 0.387·41-s + 1.56·43-s − 0.664·45-s + 1.87·47-s − 0.985·49-s − 0.421·53-s + 0.190·55-s − 0.227·57-s − 1.02·59-s + 1.26·61-s − 0.0964·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.627081927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627081927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 0.765T + 3T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 - 0.317T + 7T^{2} \) |
| 11 | \( 1 - 0.765T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 - 0.765T + 37T^{2} \) |
| 41 | \( 1 + 2.48T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 - 7.97T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.362485749124125658117027836453, −7.73088085719462712231039445417, −6.75719963531085648724929364175, −6.15071553855150426440337955187, −5.42834479521184643996252930488, −4.66322816599808284716855633886, −3.65539540866603663930205813451, −2.74272453226169346788245274437, −2.12394319276537233598388895424, −0.890705469975402738292906658611,
0.890705469975402738292906658611, 2.12394319276537233598388895424, 2.74272453226169346788245274437, 3.65539540866603663930205813451, 4.66322816599808284716855633886, 5.42834479521184643996252930488, 6.15071553855150426440337955187, 6.75719963531085648724929364175, 7.73088085719462712231039445417, 8.362485749124125658117027836453
