| L(s) = 1 | − 2.51e4·2-s + 3.66e8·4-s + 6.78e11·7-s − 2.47e12·8-s + 2.28e13·9-s + 5.68e14·11-s − 1.70e16·14-s − 3.60e16·16-s − 5.76e17·18-s − 1.43e19·22-s − 1.83e19·23-s + 3.72e19·25-s + 2.48e20·28-s + 2.94e20·29-s + 1.57e21·32-s + 8.38e21·36-s − 5.39e21·37-s + 8.74e22·43-s + 2.08e23·44-s + 4.63e23·46-s + 4.59e23·49-s − 9.38e23·50-s − 1.93e24·53-s − 1.67e24·56-s − 7.41e24·58-s + 1.55e25·63-s − 2.99e25·64-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.36·4-s + 7-s − 0.562·8-s + 9-s + 1.49·11-s − 1.53·14-s − 0.500·16-s − 1.53·18-s − 2.30·22-s − 1.58·23-s + 25-s + 1.36·28-s + 0.989·29-s + 1.33·32-s + 1.36·36-s − 0.598·37-s + 1.18·43-s + 2.04·44-s + 2.44·46-s + 49-s − 1.53·50-s − 1.40·53-s − 0.562·56-s − 1.52·58-s + 63-s − 1.54·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{29}{2})\) |
\(\approx\) |
\(1.273913219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273913219\) |
| \(L(15)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{14} T \) |
| good | 2 | \( 1 + 25199 T + p^{28} T^{2} \) |
| 3 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 5 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 11 | \( 1 - 568192759333714 T + p^{28} T^{2} \) |
| 13 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 17 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 19 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 23 | \( 1 + 18394326757532094494 T + p^{28} T^{2} \) |
| 29 | \( 1 - \)\(29\!\cdots\!94\)\( T + p^{28} T^{2} \) |
| 31 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 37 | \( 1 + \)\(53\!\cdots\!14\)\( T + p^{28} T^{2} \) |
| 41 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 43 | \( 1 - \)\(87\!\cdots\!06\)\( T + p^{28} T^{2} \) |
| 47 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 53 | \( 1 + \)\(19\!\cdots\!62\)\( T + p^{28} T^{2} \) |
| 59 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 61 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 67 | \( 1 - \)\(59\!\cdots\!66\)\( T + p^{28} T^{2} \) |
| 71 | \( 1 + \)\(14\!\cdots\!46\)\( T + p^{28} T^{2} \) |
| 73 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 79 | \( 1 - \)\(64\!\cdots\!34\)\( T + p^{28} T^{2} \) |
| 83 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 89 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 97 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
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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
−15.93804195548095678425704110342, −14.24969343257818669202313487775, −11.97762873468200571247743591567, −10.58631928583297483171713879117, −9.295654392779987947191608508145, −8.040491481529012341994375743931, −6.72823331811219035319726139558, −4.33094946849864392265524210183, −1.82768771001228429961833155644, −0.952380515168626715889351580275,
0.952380515168626715889351580275, 1.82768771001228429961833155644, 4.33094946849864392265524210183, 6.72823331811219035319726139558, 8.040491481529012341994375743931, 9.295654392779987947191608508145, 10.58631928583297483171713879117, 11.97762873468200571247743591567, 14.24969343257818669202313487775, 15.93804195548095678425704110342
