| L(s) = 1 | + 3.61e6·2-s + 8.68e12·4-s − 5.58e17·7-s + 1.55e19·8-s + 1.09e20·9-s + 6.86e21·11-s − 2.02e24·14-s + 1.79e25·16-s + 3.95e26·18-s + 2.48e28·22-s + 6.56e28·23-s + 2.27e29·25-s − 4.85e30·28-s − 9.48e28·29-s − 3.33e30·32-s + 9.50e32·36-s + 1.61e33·37-s + 6.56e33·43-s + 5.96e34·44-s + 2.37e35·46-s + 3.11e35·49-s + 8.22e35·50-s − 3.00e36·53-s − 8.67e36·56-s − 3.43e35·58-s − 6.11e37·63-s − 9.10e37·64-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.97·4-s − 7-s + 1.68·8-s + 9-s + 0.927·11-s − 1.72·14-s + 0.928·16-s + 1.72·18-s + 1.60·22-s + 1.66·23-s + 25-s − 1.97·28-s − 0.0184·29-s − 0.0823·32-s + 1.97·36-s + 1.89·37-s + 0.327·43-s + 1.83·44-s + 2.86·46-s + 49-s + 1.72·50-s − 1.85·53-s − 1.68·56-s − 0.0318·58-s − 63-s − 1.07·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{43}{2})\) |
\(\approx\) |
\(6.983636650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.983636650\) |
| \(L(22)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{21} T \) |
| good | 2 | \( 1 - 3617721 T + p^{42} T^{2} \) |
| 3 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 5 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 11 | \( 1 - \)\(68\!\cdots\!22\)\( T + p^{42} T^{2} \) |
| 13 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 17 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 19 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 23 | \( 1 - \)\(65\!\cdots\!54\)\( T + p^{42} T^{2} \) |
| 29 | \( 1 + \)\(94\!\cdots\!42\)\( T + p^{42} T^{2} \) |
| 31 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 37 | \( 1 - \)\(16\!\cdots\!26\)\( T + p^{42} T^{2} \) |
| 41 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 43 | \( 1 - \)\(65\!\cdots\!14\)\( T + p^{42} T^{2} \) |
| 47 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 53 | \( 1 + \)\(30\!\cdots\!06\)\( T + p^{42} T^{2} \) |
| 59 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 61 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 67 | \( 1 - \)\(26\!\cdots\!66\)\( T + p^{42} T^{2} \) |
| 71 | \( 1 - \)\(99\!\cdots\!42\)\( T + p^{42} T^{2} \) |
| 73 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 79 | \( 1 + \)\(10\!\cdots\!42\)\( T + p^{42} T^{2} \) |
| 83 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 89 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 97 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
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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
−13.13857319354967428202378305676, −12.53015866304110863217801863415, −11.08681847571527007839499066368, −9.438433938713728444683278226222, −7.06410348779888495611619382620, −6.29340734840380585715079505594, −4.80746584408947471146416361925, −3.77288301332309471651001686545, −2.73293591888556515919811576468, −1.12962827340926333978689261832,
1.12962827340926333978689261832, 2.73293591888556515919811576468, 3.77288301332309471651001686545, 4.80746584408947471146416361925, 6.29340734840380585715079505594, 7.06410348779888495611619382620, 9.438433938713728444683278226222, 11.08681847571527007839499066368, 12.53015866304110863217801863415, 13.13857319354967428202378305676
