| L(s) = 1 | − 79·2-s − 14·3-s + 2.14e3·4-s + 1.10e3·6-s + 1.54e5·8-s − 5.31e5·9-s − 3.00e4·12-s − 8.48e6·13-s − 2.09e7·16-s + 4.19e7·18-s + 1.48e8·23-s − 2.15e6·24-s + 2.44e8·25-s + 6.70e8·26-s + 1.48e7·27-s − 2.44e8·29-s + 1.67e9·31-s + 1.02e9·32-s − 1.13e9·36-s + 1.18e8·39-s − 7.59e9·41-s − 1.16e10·46-s + 2.06e10·47-s + 2.93e8·48-s + 1.38e10·49-s − 1.92e10·50-s − 1.81e10·52-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.0192·3-s + 0.523·4-s + 0.0237·6-s + 0.587·8-s − 0.999·9-s − 0.0100·12-s − 1.75·13-s − 1.24·16-s + 1.23·18-s + 23-s − 0.0112·24-s + 25-s + 2.16·26-s + 0.0384·27-s − 0.410·29-s + 1.88·31-s + 0.954·32-s − 0.523·36-s + 0.0337·39-s − 1.59·41-s − 1.23·46-s + 1.91·47-s + 0.0239·48-s + 49-s − 1.23·50-s − 0.920·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.6403274988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6403274988\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 + 79 T + p^{12} T^{2} \) |
| 3 | \( 1 + 14 T + p^{12} T^{2} \) |
| 5 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( 1 + 8482894 T + p^{12} T^{2} \) |
| 17 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( 1 + 244330126 T + p^{12} T^{2} \) |
| 31 | \( 1 - 1677025154 T + p^{12} T^{2} \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( 1 + 7596282526 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( 1 - 20606906306 T + p^{12} T^{2} \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 19874527918 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( 1 - 188893891874 T + p^{12} T^{2} \) |
| 73 | \( 1 - 223017449186 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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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
−15.16226956191892785919605429048, −13.84486779779488752036306715410, −12.11103814501943846851616070572, −10.70723777757242635737604258287, −9.498733427925661399585529516360, −8.368755551809186812534432651154, −7.04299524026610931367120786338, −4.96442125224735819972878972106, −2.54909073642643820759890999845, −0.64136058358020058492514381692,
0.64136058358020058492514381692, 2.54909073642643820759890999845, 4.96442125224735819972878972106, 7.04299524026610931367120786338, 8.368755551809186812534432651154, 9.498733427925661399585529516360, 10.70723777757242635737604258287, 12.11103814501943846851616070572, 13.84486779779488752036306715410, 15.16226956191892785919605429048
