| L(s) = 1 | + 1.61·2-s − 2.35·3-s + 0.607·4-s + 0.981·5-s − 3.80·6-s − 0.0820·7-s − 2.24·8-s + 2.54·9-s + 1.58·10-s − 0.0155·11-s − 1.43·12-s − 4.49·13-s − 0.132·14-s − 2.31·15-s − 4.84·16-s + 0.0424·17-s + 4.11·18-s − 7.23·19-s + 0.595·20-s + 0.193·21-s − 0.0250·22-s + 4.78·23-s + 5.29·24-s − 4.03·25-s − 7.25·26-s + 1.06·27-s − 0.0497·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.36·3-s + 0.303·4-s + 0.438·5-s − 1.55·6-s − 0.0309·7-s − 0.795·8-s + 0.849·9-s + 0.501·10-s − 0.00467·11-s − 0.412·12-s − 1.24·13-s − 0.0353·14-s − 0.596·15-s − 1.21·16-s + 0.0103·17-s + 0.970·18-s − 1.65·19-s + 0.133·20-s + 0.0421·21-s − 0.00534·22-s + 0.998·23-s + 1.08·24-s − 0.807·25-s − 1.42·26-s + 0.204·27-s − 0.00940·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 0.981T + 5T^{2} \) |
| 7 | \( 1 + 0.0820T + 7T^{2} \) |
| 11 | \( 1 + 0.0155T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 0.0424T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 7.53T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 - 2.90T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−10.39087187348498588604496891287, −9.581837174986175068255932828561, −8.440829664653567213488457951890, −6.89524501130398005758433760062, −6.35737933468548831259544274059, −5.29464392667257592709577080929, −4.96857695947055515553031895368, −3.79790943049637903319398987654, −2.28176461960288522958136795105, 0,
2.28176461960288522958136795105, 3.79790943049637903319398987654, 4.96857695947055515553031895368, 5.29464392667257592709577080929, 6.35737933468548831259544274059, 6.89524501130398005758433760062, 8.440829664653567213488457951890, 9.581837174986175068255932828561, 10.39087187348498588604496891287
