| L(s) = 1 | − 4.88e9·2-s − 4.99e19·4-s − 7.73e27·7-s + 6.04e29·8-s + 3.09e31·9-s − 1.41e34·11-s + 3.77e37·14-s + 7.32e38·16-s − 1.50e41·18-s + 6.90e43·22-s + 1.11e45·23-s + 1.35e46·25-s + 3.86e47·28-s − 3.52e48·29-s − 4.81e49·32-s − 1.54e51·36-s + 9.79e51·37-s − 1.04e54·43-s + 7.05e53·44-s − 5.45e54·46-s + 5.97e55·49-s − 6.61e55·50-s − 1.52e57·53-s − 4.67e57·56-s + 1.72e58·58-s − 2.38e59·63-s + 1.81e59·64-s + ⋯ |
| L(s) = 1 | − 0.568·2-s − 0.676·4-s − 7-s + 0.953·8-s + 9-s − 0.608·11-s + 0.568·14-s + 0.134·16-s − 0.568·18-s + 0.346·22-s + 1.29·23-s + 25-s + 0.676·28-s − 1.94·29-s − 1.02·32-s − 0.676·36-s + 1.73·37-s − 1.29·43-s + 0.411·44-s − 0.733·46-s + 49-s − 0.568·50-s − 1.91·53-s − 0.953·56-s + 1.10·58-s − 63-s + 0.451·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{67}{2})\) |
\(\approx\) |
\(0.9123232873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9123232873\) |
| \(L(34)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{33} T \) |
| good | 2 | \( 1 + 4884385959 T + p^{66} T^{2} \) |
| 3 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 5 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 11 | \( 1 + \)\(14\!\cdots\!38\)\( T + p^{66} T^{2} \) |
| 13 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 17 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 19 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 23 | \( 1 - \)\(11\!\cdots\!34\)\( T + p^{66} T^{2} \) |
| 29 | \( 1 + \)\(35\!\cdots\!22\)\( T + p^{66} T^{2} \) |
| 31 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 37 | \( 1 - \)\(97\!\cdots\!06\)\( T + p^{66} T^{2} \) |
| 41 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 43 | \( 1 + \)\(10\!\cdots\!86\)\( T + p^{66} T^{2} \) |
| 47 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 53 | \( 1 + \)\(15\!\cdots\!46\)\( T + p^{66} T^{2} \) |
| 59 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 61 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 67 | \( 1 - \)\(30\!\cdots\!26\)\( T + p^{66} T^{2} \) |
| 71 | \( 1 + \)\(15\!\cdots\!78\)\( T + p^{66} T^{2} \) |
| 73 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 79 | \( 1 + \)\(68\!\cdots\!22\)\( T + p^{66} T^{2} \) |
| 83 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 89 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
| 97 | \( ( 1 - p^{33} T )( 1 + p^{33} T ) \) |
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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
−10.90730401145556568009798231635, −9.840402780689557441899367586146, −9.149433979131365712186021419431, −7.78818983650860127500620871582, −6.79310257487577436808658784162, −5.27324026868943085777190433639, −4.18124100984963782590587361179, −3.01801499147486761826246519448, −1.49523090806880073956922554130, −0.46894079401630117636010599483,
0.46894079401630117636010599483, 1.49523090806880073956922554130, 3.01801499147486761826246519448, 4.18124100984963782590587361179, 5.27324026868943085777190433639, 6.79310257487577436808658784162, 7.78818983650860127500620871582, 9.149433979131365712186021419431, 9.840402780689557441899367586146, 10.90730401145556568009798231635
