| L(s) = 1 | − 8.47e11·3-s − 1.14e21·7-s + 7.17e23·9-s − 1.20e27·13-s − 2.14e30·19-s + 9.72e32·21-s + 8.88e34·25-s − 6.08e35·27-s − 3.41e37·31-s + 1.39e39·37-s + 1.02e39·39-s − 2.51e40·43-s − 4.81e41·49-s + 1.82e42·57-s − 4.74e44·61-s − 8.23e44·63-s − 2.94e45·67-s − 7.56e46·73-s − 7.52e46·75-s − 1.90e47·79-s + 5.15e47·81-s + 1.38e48·91-s + 2.89e49·93-s + 2.38e49·97-s − 2.76e50·103-s − 1.44e50·109-s − 1.18e51·111-s + ⋯ |
| L(s) = 1 | − 3-s − 0.855·7-s + 9-s − 0.170·13-s − 0.0230·19-s + 0.855·21-s + 25-s − 27-s − 1.77·31-s + 0.868·37-s + 0.170·39-s − 0.366·43-s − 0.267·49-s + 0.0230·57-s − 1.10·61-s − 0.855·63-s − 0.657·67-s − 1.97·73-s − 75-s − 0.690·79-s + 81-s + 0.146·91-s + 1.77·93-s + 0.509·97-s − 1.32·103-s − 0.167·109-s − 0.868·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(51-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+25) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{51}{2})\) |
\(\approx\) |
\(0.7966899825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7966899825\) |
| \(L(26)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{25} T \) |
| good | 5 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 7 | \( 1 + \)\(11\!\cdots\!18\)\( T + p^{50} T^{2} \) |
| 11 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 13 | \( 1 + \)\(12\!\cdots\!82\)\( T + p^{50} T^{2} \) |
| 17 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 19 | \( 1 + \)\(21\!\cdots\!74\)\( T + p^{50} T^{2} \) |
| 23 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 29 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 31 | \( 1 + \)\(34\!\cdots\!26\)\( T + p^{50} T^{2} \) |
| 37 | \( 1 - \)\(13\!\cdots\!86\)\( T + p^{50} T^{2} \) |
| 41 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 43 | \( 1 + \)\(25\!\cdots\!82\)\( T + p^{50} T^{2} \) |
| 47 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 53 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 59 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 61 | \( 1 + \)\(47\!\cdots\!26\)\( T + p^{50} T^{2} \) |
| 67 | \( 1 + \)\(29\!\cdots\!18\)\( T + p^{50} T^{2} \) |
| 71 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 73 | \( 1 + \)\(75\!\cdots\!86\)\( T + p^{50} T^{2} \) |
| 79 | \( 1 + \)\(19\!\cdots\!02\)\( T + p^{50} T^{2} \) |
| 83 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 89 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 97 | \( 1 - \)\(23\!\cdots\!82\)\( T + p^{50} T^{2} \) |
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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
−11.22020889573260139225365297756, −10.20120449618955205434233079776, −9.129211700574231642450965199146, −7.42746181181000495731849318406, −6.46625267231852767256372481027, −5.48609049759588480383068999781, −4.32246112996044208807580406473, −3.07508941095307872261851464916, −1.59202604693371140049001881746, −0.40431933893126153832011096965,
0.40431933893126153832011096965, 1.59202604693371140049001881746, 3.07508941095307872261851464916, 4.32246112996044208807580406473, 5.48609049759588480383068999781, 6.46625267231852767256372481027, 7.42746181181000495731849318406, 9.129211700574231642450965199146, 10.20120449618955205434233079776, 11.22020889573260139225365297756
