| L(s) = 1 | + 7.38·2-s + 13.5·3-s + 22.5·4-s − 1.22·5-s + 100.·6-s − 233.·7-s − 69.9·8-s − 58.4·9-s − 9.01·10-s + 68.2·11-s + 306.·12-s + 411.·13-s − 1.72e3·14-s − 16.5·15-s − 1.23e3·16-s − 587.·17-s − 431.·18-s + 1.02e3·19-s − 27.5·20-s − 3.17e3·21-s + 503.·22-s − 2.96e3·23-s − 949.·24-s − 3.12e3·25-s + 3.03e3·26-s − 4.09e3·27-s − 5.26e3·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.871·3-s + 0.704·4-s − 0.0218·5-s + 1.13·6-s − 1.80·7-s − 0.386·8-s − 0.240·9-s − 0.0285·10-s + 0.169·11-s + 0.613·12-s + 0.675·13-s − 2.35·14-s − 0.0190·15-s − 1.20·16-s − 0.493·17-s − 0.313·18-s + 0.653·19-s − 0.0153·20-s − 1.57·21-s + 0.221·22-s − 1.16·23-s − 0.336·24-s − 0.999·25-s + 0.881·26-s − 1.08·27-s − 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 229 | \( 1 + 5.24e4T \) |
| good | 2 | \( 1 - 7.38T + 32T^{2} \) |
| 3 | \( 1 - 13.5T + 243T^{2} \) |
| 5 | \( 1 + 1.22T + 3.12e3T^{2} \) |
| 7 | \( 1 + 233.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 68.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 411.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 587.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 889.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.22e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 875.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.07e5T + 8.58e9T^{2} \) |
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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
−11.13508043864095980929780141690, −9.576290224916988785099575283776, −9.153926802666545081589067213102, −7.74658709795843403929742514090, −6.33581208615296127604549793700, −5.80478197647316919065847795333, −4.03592192955432299183058573904, −3.40492937081177006526262786420, −2.44850122881032885605738364624, 0,
2.44850122881032885605738364624, 3.40492937081177006526262786420, 4.03592192955432299183058573904, 5.80478197647316919065847795333, 6.33581208615296127604549793700, 7.74658709795843403929742514090, 9.153926802666545081589067213102, 9.576290224916988785099575283776, 11.13508043864095980929780141690
