| L(s) = 1 | − 2.04e3·2-s − 5.05e5·3-s + 4.19e6·4-s − 9.01e7·5-s + 1.03e9·6-s + 6.87e9·7-s − 8.58e9·8-s + 1.61e11·9-s + 1.84e11·10-s − 9.65e11·11-s − 2.12e12·12-s + 5.42e11·13-s − 1.40e13·14-s + 4.56e13·15-s + 1.75e13·16-s + 8.20e13·17-s − 3.31e14·18-s + 5.55e14·19-s − 3.78e14·20-s − 3.47e15·21-s + 1.97e15·22-s + 6.50e15·23-s + 4.34e15·24-s − 3.79e15·25-s − 1.11e15·26-s − 3.42e16·27-s + 2.88e16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.64·3-s + 1/2·4-s − 0.825·5-s + 1.16·6-s + 1.31·7-s − 0.353·8-s + 1.71·9-s + 0.583·10-s − 1.02·11-s − 0.824·12-s + 0.0839·13-s − 0.928·14-s + 1.36·15-s + 1/4·16-s + 0.580·17-s − 1.21·18-s + 1.09·19-s − 0.412·20-s − 2.16·21-s + 0.721·22-s + 1.42·23-s + 0.582·24-s − 0.318·25-s − 0.0593·26-s − 1.18·27-s + 0.656·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.5970484347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5970484347\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{11} T \) |
| good | 3 | \( 1 + 56212 p^{2} T + p^{23} T^{2} \) |
| 5 | \( 1 + 18027114 p T + p^{23} T^{2} \) |
| 7 | \( 1 - 140250104 p^{2} T + p^{23} T^{2} \) |
| 11 | \( 1 + 87757163508 p T + p^{23} T^{2} \) |
| 13 | \( 1 - 41719999934 p T + p^{23} T^{2} \) |
| 17 | \( 1 - 82083537265266 T + p^{23} T^{2} \) |
| 19 | \( 1 - 29249923769300 p T + p^{23} T^{2} \) |
| 23 | \( 1 - 6508638190765032 T + p^{23} T^{2} \) |
| 29 | \( 1 + 12202037915600490 T + p^{23} T^{2} \) |
| 31 | \( 1 - 119978011042749152 T + p^{23} T^{2} \) |
| 37 | \( 1 + 619510980267421234 T + p^{23} T^{2} \) |
| 41 | \( 1 + 1587735553771936038 T + p^{23} T^{2} \) |
| 43 | \( 1 - 8377717142038508132 T + p^{23} T^{2} \) |
| 47 | \( 1 - 13100457020745462096 T + p^{23} T^{2} \) |
| 53 | \( 1 - 41795979279875033022 T + p^{23} T^{2} \) |
| 59 | \( 1 + 74383865281405054380 T + p^{23} T^{2} \) |
| 61 | \( 1 + \)\(27\!\cdots\!98\)\( T + p^{23} T^{2} \) |
| 67 | \( 1 - \)\(17\!\cdots\!76\)\( T + p^{23} T^{2} \) |
| 71 | \( 1 + \)\(27\!\cdots\!68\)\( T + p^{23} T^{2} \) |
| 73 | \( 1 - \)\(43\!\cdots\!62\)\( T + p^{23} T^{2} \) |
| 79 | \( 1 - \)\(35\!\cdots\!40\)\( T + p^{23} T^{2} \) |
| 83 | \( 1 + \)\(22\!\cdots\!28\)\( T + p^{23} T^{2} \) |
| 89 | \( 1 - \)\(33\!\cdots\!10\)\( T + p^{23} T^{2} \) |
| 97 | \( 1 - \)\(92\!\cdots\!06\)\( T + p^{23} T^{2} \) |
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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
−23.12439400803997885876267624093, −21.03437751182473486108765403991, −18.53186568121701896139140339616, −17.29187302279347886827786807155, −15.70305834539754169581032458002, −11.88364652489768543939994497968, −10.79536396694016956551931686803, −7.61324924249376789382501412424, −5.17303268368456456207718330014, −0.846104671131657956648477724994,
0.846104671131657956648477724994, 5.17303268368456456207718330014, 7.61324924249376789382501412424, 10.79536396694016956551931686803, 11.88364652489768543939994497968, 15.70305834539754169581032458002, 17.29187302279347886827786807155, 18.53186568121701896139140339616, 21.03437751182473486108765403991, 23.12439400803997885876267624093
