| L(s) = 1 | + 1.67e7·2-s − 1.68e11·3-s + 2.81e14·4-s − 2.81e18·6-s + 4.72e21·8-s − 5.15e22·9-s − 1.30e25·11-s − 4.73e25·12-s + 7.92e28·16-s + 1.06e29·17-s − 8.64e29·18-s + 1.12e30·19-s − 2.18e32·22-s − 7.93e32·24-s + 3.55e33·25-s + 2.20e34·27-s + 1.32e36·32-s + 2.19e36·33-s + 1.79e36·34-s − 1.45e37·36-s + 1.89e37·38-s − 1.58e38·41-s − 2.26e39·43-s − 3.67e39·44-s − 1.33e40·48-s + 3.67e40·49-s + 5.96e40·50-s + ⋯ |
| L(s) = 1 | + 2-s − 0.595·3-s + 4-s − 0.595·6-s + 8-s − 0.645·9-s − 1.32·11-s − 0.595·12-s + 16-s + 0.315·17-s − 0.645·18-s + 0.230·19-s − 1.32·22-s − 0.595·24-s + 25-s + 0.979·27-s + 32-s + 0.788·33-s + 0.315·34-s − 0.645·36-s + 0.230·38-s − 0.311·41-s − 1.41·43-s − 1.32·44-s − 0.595·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(49-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+24) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{49}{2})\) |
\(\approx\) |
\(3.067006828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.067006828\) |
| \(L(25)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{24} T \) |
| good | 3 | \( 1 + 168062386238 T + p^{48} T^{2} \) |
| 5 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 7 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 11 | \( 1 + \)\(13\!\cdots\!18\)\( T + p^{48} T^{2} \) |
| 13 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 17 | \( 1 - \)\(10\!\cdots\!42\)\( T + p^{48} T^{2} \) |
| 19 | \( 1 - \)\(11\!\cdots\!42\)\( T + p^{48} T^{2} \) |
| 23 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 29 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 31 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 37 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 41 | \( 1 + \)\(15\!\cdots\!78\)\( T + p^{48} T^{2} \) |
| 43 | \( 1 + \)\(22\!\cdots\!98\)\( T + p^{48} T^{2} \) |
| 47 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 53 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 59 | \( 1 - \)\(58\!\cdots\!22\)\( T + p^{48} T^{2} \) |
| 61 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 67 | \( 1 - \)\(69\!\cdots\!42\)\( T + p^{48} T^{2} \) |
| 71 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 73 | \( 1 - \)\(83\!\cdots\!82\)\( T + p^{48} T^{2} \) |
| 79 | \( ( 1 - p^{24} T )( 1 + p^{24} T ) \) |
| 83 | \( 1 - \)\(83\!\cdots\!42\)\( T + p^{48} T^{2} \) |
| 89 | \( 1 + \)\(56\!\cdots\!18\)\( T + p^{48} T^{2} \) |
| 97 | \( 1 - \)\(87\!\cdots\!62\)\( T + p^{48} T^{2} \) |
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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
−12.38079433555259897142460864492, −11.26966968576721348869750105763, −10.29086419508766991189140531366, −8.211945127214893889983402342768, −6.85856510485427112683756726454, −5.59796448574006454441430052309, −4.92198360795573437814350506490, −3.33117950880412348358771783019, −2.31179251172166240717505276534, −0.70788159369601408813213209801,
0.70788159369601408813213209801, 2.31179251172166240717505276534, 3.33117950880412348358771783019, 4.92198360795573437814350506490, 5.59796448574006454441430052309, 6.85856510485427112683756726454, 8.211945127214893889983402342768, 10.29086419508766991189140531366, 11.26966968576721348869750105763, 12.38079433555259897142460864492
