| L(s) = 1 | + 9.81e7·2-s − 6.24e16·4-s + 4.59e23·7-s − 1.31e25·8-s + 5.23e26·9-s + 3.44e28·11-s + 4.51e31·14-s + 3.20e33·16-s + 5.13e34·18-s + 3.37e36·22-s + 6.95e37·23-s + 1.38e39·25-s − 2.87e40·28-s − 9.04e40·29-s + 1.26e42·32-s − 3.26e43·36-s − 1.33e44·37-s − 3.27e45·43-s − 2.14e45·44-s + 6.82e45·46-s + 2.11e47·49-s + 1.36e47·50-s − 5.69e46·53-s − 6.06e48·56-s − 8.87e48·58-s + 2.40e50·63-s − 1.06e50·64-s + ⋯ |
| L(s) = 1 | + 0.365·2-s − 0.866·4-s + 7-s − 0.682·8-s + 9-s + 0.238·11-s + 0.365·14-s + 0.617·16-s + 0.365·18-s + 0.0872·22-s + 0.517·23-s + 25-s − 0.866·28-s − 1.02·29-s + 0.907·32-s − 0.866·36-s − 1.64·37-s − 0.599·43-s − 0.206·44-s + 0.189·46-s + 49-s + 0.365·50-s − 0.0299·53-s − 0.682·56-s − 0.373·58-s + 63-s − 0.285·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(57-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+28) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{57}{2})\) |
\(\approx\) |
\(2.691063296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691063296\) |
| \(L(29)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{28} T \) |
| good | 2 | \( 1 - 98118689 T + p^{56} T^{2} \) |
| 3 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 5 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 11 | \( 1 - \)\(34\!\cdots\!34\)\( T + p^{56} T^{2} \) |
| 13 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 17 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 19 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 23 | \( 1 - \)\(69\!\cdots\!74\)\( T + p^{56} T^{2} \) |
| 29 | \( 1 + \)\(90\!\cdots\!86\)\( T + p^{56} T^{2} \) |
| 31 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 37 | \( 1 + \)\(13\!\cdots\!46\)\( T + p^{56} T^{2} \) |
| 41 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 43 | \( 1 + \)\(32\!\cdots\!66\)\( T + p^{56} T^{2} \) |
| 47 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 53 | \( 1 + \)\(56\!\cdots\!78\)\( T + p^{56} T^{2} \) |
| 59 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 61 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 67 | \( 1 - \)\(83\!\cdots\!74\)\( T + p^{56} T^{2} \) |
| 71 | \( 1 - \)\(78\!\cdots\!94\)\( T + p^{56} T^{2} \) |
| 73 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 79 | \( 1 - \)\(14\!\cdots\!34\)\( T + p^{56} T^{2} \) |
| 83 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 89 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
| 97 | \( ( 1 - p^{28} T )( 1 + p^{28} T ) \) |
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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
−12.17796738075393806019582021312, −10.72445969063641574394481608253, −9.396682758796202351692448511406, −8.300054467971678704872937419408, −6.98299051945250417710894800270, −5.34020775366084687700796475512, −4.53812617102688119330487380319, −3.49317666435957468371173845186, −1.79368835236140323353096823221, −0.74287433347432962124817826957,
0.74287433347432962124817826957, 1.79368835236140323353096823221, 3.49317666435957468371173845186, 4.53812617102688119330487380319, 5.34020775366084687700796475512, 6.98299051945250417710894800270, 8.300054467971678704872937419408, 9.396682758796202351692448511406, 10.72445969063641574394481608253, 12.17796738075393806019582021312
