| L(s) = 1 | − 1.16e11·3-s + 7.03e13·4-s + 1.77e16·5-s + 4.68e21·9-s − 8.95e23·11-s − 8.19e24·12-s − 2.06e27·15-s + 4.95e27·16-s + 1.24e30·20-s − 3.59e31·23-s + 1.71e32·25-s + 4.86e32·27-s + 3.36e34·31-s + 1.04e35·33-s + 3.29e35·36-s + 2.33e36·37-s − 6.30e37·44-s + 8.29e37·45-s − 1.88e38·47-s − 5.76e38·48-s + 7.49e38·49-s − 7.03e39·53-s − 1.58e40·55-s − 8.94e40·59-s − 1.45e41·60-s + 3.48e41·64-s + 4.07e41·67-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 4-s + 1.48·5-s + 0.528·9-s − 11-s − 1.23·12-s − 1.83·15-s + 16-s + 1.48·20-s − 1.72·23-s + 1.20·25-s + 0.582·27-s + 1.68·31-s + 1.23·33-s + 0.528·36-s + 1.99·37-s − 44-s + 0.785·45-s − 0.654·47-s − 1.23·48-s + 49-s − 1.54·53-s − 1.48·55-s − 1.66·59-s − 1.83·60-s + 64-s + 0.407·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(47-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+23) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{47}{2})\) |
\(\approx\) |
\(2.326364664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326364664\) |
| \(L(24)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{23} T \) |
| good | 2 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 3 | \( 1 + 116391051965 T + p^{46} T^{2} \) |
| 5 | \( 1 - 17717696898466199 T + p^{46} T^{2} \) |
| 7 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 13 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 17 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 19 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 23 | \( 1 + \)\(35\!\cdots\!45\)\( T + p^{46} T^{2} \) |
| 29 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 31 | \( 1 - \)\(33\!\cdots\!43\)\( T + p^{46} T^{2} \) |
| 37 | \( 1 - \)\(23\!\cdots\!75\)\( T + p^{46} T^{2} \) |
| 41 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 43 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 47 | \( 1 + \)\(18\!\cdots\!50\)\( T + p^{46} T^{2} \) |
| 53 | \( 1 + \)\(70\!\cdots\!10\)\( T + p^{46} T^{2} \) |
| 59 | \( 1 + \)\(89\!\cdots\!33\)\( T + p^{46} T^{2} \) |
| 61 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 67 | \( 1 - \)\(40\!\cdots\!55\)\( T + p^{46} T^{2} \) |
| 71 | \( 1 - \)\(28\!\cdots\!47\)\( T + p^{46} T^{2} \) |
| 73 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 79 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 83 | \( ( 1 - p^{23} T )( 1 + p^{23} T ) \) |
| 89 | \( 1 - \)\(87\!\cdots\!63\)\( T + p^{46} T^{2} \) |
| 97 | \( 1 - \)\(70\!\cdots\!35\)\( T + p^{46} T^{2} \) |
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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
−11.64227515304636857749395774992, −10.55984381212619768788350935840, −9.865203379513574271642701581261, −7.86520692581667772504684039491, −6.23407282552857325673268054217, −6.03769221989746554911660679740, −4.86651938670274545623459595327, −2.76072361604702784562035995375, −1.89027081260852528600114790575, −0.71328557353953831375300623448,
0.71328557353953831375300623448, 1.89027081260852528600114790575, 2.76072361604702784562035995375, 4.86651938670274545623459595327, 6.03769221989746554911660679740, 6.23407282552857325673268054217, 7.86520692581667772504684039491, 9.865203379513574271642701581261, 10.55984381212619768788350935840, 11.64227515304636857749395774992
