| L(s) = 1 | + 0.879·2-s + 0.532·3-s − 1.22·4-s − 2.53·5-s + 0.467·6-s − 1.87·7-s − 2.83·8-s − 2.71·9-s − 2.22·10-s + 3.41·11-s − 0.652·12-s − 5.29·13-s − 1.65·14-s − 1.34·15-s − 0.0418·16-s + 1.65·17-s − 2.38·18-s + 3.10·20-s − 21-s + 2.99·22-s + 1.75·23-s − 1.50·24-s + 1.41·25-s − 4.65·26-s − 3.04·27-s + 2.30·28-s − 3.46·29-s + ⋯ |
| L(s) = 1 | + 0.621·2-s + 0.307·3-s − 0.613·4-s − 1.13·5-s + 0.191·6-s − 0.710·7-s − 1.00·8-s − 0.905·9-s − 0.704·10-s + 1.02·11-s − 0.188·12-s − 1.46·13-s − 0.441·14-s − 0.347·15-s − 0.0104·16-s + 0.400·17-s − 0.563·18-s + 0.694·20-s − 0.218·21-s + 0.639·22-s + 0.366·23-s − 0.308·24-s + 0.282·25-s − 0.912·26-s − 0.585·27-s + 0.435·28-s − 0.643·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 \) |
| good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 3 | \( 1 - 0.532T + 3T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 0.837T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 - 0.716T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 + 6.96T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−11.34995970496355596885635618897, −9.818448745372216294856213540657, −9.144966515604289972263318379324, −8.212829765642751559374590817432, −7.19206638251505883684715728846, −5.98213818070361350857986914401, −4.80565508087246501286513044331, −3.77970232341950343517802239948, −2.96754160901964914621753360684, 0,
2.96754160901964914621753360684, 3.77970232341950343517802239948, 4.80565508087246501286513044331, 5.98213818070361350857986914401, 7.19206638251505883684715728846, 8.212829765642751559374590817432, 9.144966515604289972263318379324, 9.818448745372216294856213540657, 11.34995970496355596885635618897
