| L(s) = 1 | − 43.6·2-s − 243·3-s − 142.·4-s + 7.12e3·5-s + 1.06e4·6-s + 5.03e4·7-s + 9.56e4·8-s + 5.90e4·9-s − 3.11e5·10-s + 3.65e5·11-s + 3.45e4·12-s + 6.75e5·13-s − 2.19e6·14-s − 1.73e6·15-s − 3.88e6·16-s + 5.75e6·17-s − 2.57e6·18-s + 6.02e6·19-s − 1.01e6·20-s − 1.22e7·21-s − 1.59e7·22-s − 3.76e7·23-s − 2.32e7·24-s + 1.92e6·25-s − 2.94e7·26-s − 1.43e7·27-s − 7.14e6·28-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.577·3-s − 0.0693·4-s + 1.01·5-s + 0.556·6-s + 1.13·7-s + 1.03·8-s + 0.333·9-s − 0.983·10-s + 0.683·11-s + 0.0400·12-s + 0.504·13-s − 1.09·14-s − 0.588·15-s − 0.925·16-s + 0.983·17-s − 0.321·18-s + 0.558·19-s − 0.0707·20-s − 0.653·21-s − 0.659·22-s − 1.22·23-s − 0.595·24-s + 0.0395·25-s − 0.486·26-s − 0.192·27-s − 0.0785·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 43.6T + 2.04e3T^{2} \) |
| 5 | \( 1 - 7.12e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.03e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.65e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 6.75e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.75e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.02e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.76e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.23e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 3.05e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.58e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.12e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.10e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.34e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.46e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 2.87e8T + 4.35e19T^{2} \) |
| 67 | \( 1 - 6.29e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.39e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.18e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.00e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 8.58e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.54e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.20e10T + 7.15e21T^{2} \) |
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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
−9.978671725754553581959061932006, −9.346915262629965641106380531837, −8.239539171201266322383112587944, −7.34619480532828563760643397217, −5.93386351105562095756413467677, −5.13538167729612863734370754475, −3.86994130079126221844779874438, −1.68512780134285768721516329667, −1.43832344459819988523119572844, 0,
1.43832344459819988523119572844, 1.68512780134285768721516329667, 3.86994130079126221844779874438, 5.13538167729612863734370754475, 5.93386351105562095756413467677, 7.34619480532828563760643397217, 8.239539171201266322383112587944, 9.346915262629965641106380531837, 9.978671725754553581959061932006
