| L(s) = 1 | + 4.16·2-s − 17.5·3-s − 14.6·4-s + 32.5·5-s − 72.9·6-s − 220.·7-s − 194.·8-s + 64.1·9-s + 135.·10-s − 85.7·11-s + 257.·12-s + 1.03e3·13-s − 919.·14-s − 570.·15-s − 339.·16-s − 313.·17-s + 267.·18-s + 458.·19-s − 477.·20-s + 3.86e3·21-s − 356.·22-s − 3.44e3·23-s + 3.40e3·24-s − 2.06e3·25-s + 4.30e3·26-s + 3.13e3·27-s + 3.24e3·28-s + ⋯ |
| L(s) = 1 | + 0.735·2-s − 1.12·3-s − 0.458·4-s + 0.582·5-s − 0.827·6-s − 1.70·7-s − 1.07·8-s + 0.264·9-s + 0.428·10-s − 0.213·11-s + 0.515·12-s + 1.69·13-s − 1.25·14-s − 0.654·15-s − 0.331·16-s − 0.262·17-s + 0.194·18-s + 0.291·19-s − 0.267·20-s + 1.91·21-s − 0.157·22-s − 1.35·23-s + 1.20·24-s − 0.660·25-s + 1.24·26-s + 0.827·27-s + 0.781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + 841T \) |
| good | 2 | \( 1 - 4.16T + 32T^{2} \) |
| 3 | \( 1 + 17.5T + 243T^{2} \) |
| 5 | \( 1 - 32.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 220.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 313.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 458.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.44e3T + 6.43e6T^{2} \) |
| 31 | \( 1 + 7.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 152.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.19e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.36e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.66891952819415702992310901843, −13.77335358258133985438236697668, −13.08940155189162677732128584854, −11.95045393567753199675588338820, −10.37312581913369923034260356122, −9.072611988000810662112591225342, −6.28412900245602665615722872437, −5.70081226594538631404387762560, −3.63547011652282956089496635579, 0,
3.63547011652282956089496635579, 5.70081226594538631404387762560, 6.28412900245602665615722872437, 9.072611988000810662112591225342, 10.37312581913369923034260356122, 11.95045393567753199675588338820, 13.08940155189162677732128584854, 13.77335358258133985438236697668, 15.66891952819415702992310901843
