| L(s) = 1 | − 1.17·2-s − 1.77·3-s − 0.608·4-s − 1.41·5-s + 2.09·6-s + 1.49·7-s + 3.07·8-s + 0.153·9-s + 1.66·10-s − 4.42·11-s + 1.07·12-s + 13-s − 1.76·14-s + 2.51·15-s − 2.41·16-s + 7.76·17-s − 0.180·18-s − 2.09·19-s + 0.859·20-s − 2.65·21-s + 5.21·22-s + 5.13·23-s − 5.46·24-s − 3.00·25-s − 1.17·26-s + 5.05·27-s − 0.907·28-s + ⋯ |
| L(s) = 1 | − 0.834·2-s − 1.02·3-s − 0.304·4-s − 0.632·5-s + 0.855·6-s + 0.564·7-s + 1.08·8-s + 0.0510·9-s + 0.527·10-s − 1.33·11-s + 0.311·12-s + 0.277·13-s − 0.470·14-s + 0.648·15-s − 0.603·16-s + 1.88·17-s − 0.0426·18-s − 0.479·19-s + 0.192·20-s − 0.578·21-s + 1.11·22-s + 1.07·23-s − 1.11·24-s − 0.600·25-s − 0.231·26-s + 0.972·27-s − 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 17 | \( 1 - 7.76T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.70T + 53T^{2} \) |
| 59 | \( 1 - 6.66T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 - 9.40T + 97T^{2} \) |
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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
−9.713873763896318747532291316832, −8.463561082963469012924015563837, −7.994816968121518341811756128278, −7.34016845815598885889978044139, −5.99290875552858898090004056805, −5.16406716053947912915927306834, −4.50870927691619408258674570205, −3.08142788595485815145384767713, −1.22866033851835373515191519156, 0,
1.22866033851835373515191519156, 3.08142788595485815145384767713, 4.50870927691619408258674570205, 5.16406716053947912915927306834, 5.99290875552858898090004056805, 7.34016845815598885889978044139, 7.994816968121518341811756128278, 8.463561082963469012924015563837, 9.713873763896318747532291316832
