| L(s) = 1 | + 1.50e17·3-s + 4.72e21·4-s + 2.29e30·7-s + 2.25e34·9-s + 7.08e38·12-s + 4.67e39·13-s + 2.23e43·16-s − 8.90e45·19-s + 3.44e47·21-s + 2.11e50·25-s + 3.38e51·27-s + 1.08e52·28-s + 2.41e53·31-s + 1.06e56·36-s + 5.34e56·37-s + 7.01e56·39-s − 1.26e59·43-s + 3.34e60·48-s − 1.75e60·49-s + 2.20e61·52-s − 1.33e63·57-s − 3.73e63·61-s + 5.17e64·63-s + 1.05e65·64-s − 1.02e66·67-s + 1.24e67·73-s + 3.17e67·75-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 0.866·7-s + 9-s + 12-s + 0.369·13-s + 16-s − 0.821·19-s + 0.866·21-s + 25-s + 27-s + 0.866·28-s + 0.493·31-s + 36-s + 1.87·37-s + 0.369·39-s − 1.98·43-s + 48-s − 0.249·49-s + 0.369·52-s − 0.821·57-s − 0.199·61-s + 0.866·63-s + 64-s − 1.87·67-s + 1.03·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(73-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+36) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{73}{2})\) |
\(\approx\) |
\(5.362628310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.362628310\) |
| \(L(37)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{36} T \) |
| good | 2 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 5 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 7 | \( 1 - \)\(22\!\cdots\!02\)\( T + p^{72} T^{2} \) |
| 11 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 13 | \( 1 - \)\(46\!\cdots\!82\)\( T + p^{72} T^{2} \) |
| 17 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 19 | \( 1 + \)\(89\!\cdots\!38\)\( T + p^{72} T^{2} \) |
| 23 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 29 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 31 | \( 1 - \)\(24\!\cdots\!62\)\( T + p^{72} T^{2} \) |
| 37 | \( 1 - \)\(53\!\cdots\!82\)\( T + p^{72} T^{2} \) |
| 41 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 43 | \( 1 + \)\(12\!\cdots\!98\)\( T + p^{72} T^{2} \) |
| 47 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 53 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 59 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 61 | \( 1 + \)\(37\!\cdots\!78\)\( T + p^{72} T^{2} \) |
| 67 | \( 1 + \)\(10\!\cdots\!38\)\( T + p^{72} T^{2} \) |
| 71 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 73 | \( 1 - \)\(12\!\cdots\!22\)\( T + p^{72} T^{2} \) |
| 79 | \( 1 + \)\(33\!\cdots\!58\)\( T + p^{72} T^{2} \) |
| 83 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 89 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 97 | \( 1 - \)\(62\!\cdots\!42\)\( T + p^{72} T^{2} \) |
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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
−12.96179474584450119152334625272, −11.46657023781954026215704225708, −10.26292055847372953153249896675, −8.611959740891854239685228315277, −7.64375475725282996473622446612, −6.39644072501475632616458257664, −4.60746369056557056545534484409, −3.17424955100364482901219675594, −2.10946869668896482745923119936, −1.17857105237807572092536666464,
1.17857105237807572092536666464, 2.10946869668896482745923119936, 3.17424955100364482901219675594, 4.60746369056557056545534484409, 6.39644072501475632616458257664, 7.64375475725282996473622446612, 8.611959740891854239685228315277, 10.26292055847372953153249896675, 11.46657023781954026215704225708, 12.96179474584450119152334625272
