L(s) = 1 | + 0.775·2-s + 3-s − 1.39·4-s + 3.03·5-s + 0.775·6-s − 7-s − 2.63·8-s + 9-s + 2.35·10-s − 6.14·11-s − 1.39·12-s − 0.775·14-s + 3.03·15-s + 0.750·16-s + 6.44·17-s + 0.775·18-s − 4.32·19-s − 4.24·20-s − 21-s − 4.76·22-s − 8.92·23-s − 2.63·24-s + 4.20·25-s + 27-s + 1.39·28-s − 2.44·29-s + 2.35·30-s + ⋯ |
L(s) = 1 | + 0.548·2-s + 0.577·3-s − 0.699·4-s + 1.35·5-s + 0.316·6-s − 0.377·7-s − 0.932·8-s + 0.333·9-s + 0.744·10-s − 1.85·11-s − 0.403·12-s − 0.207·14-s + 0.783·15-s + 0.187·16-s + 1.56·17-s + 0.182·18-s − 0.991·19-s − 0.948·20-s − 0.218·21-s − 1.01·22-s − 1.86·23-s − 0.538·24-s + 0.841·25-s + 0.192·27-s + 0.264·28-s − 0.454·29-s + 0.429·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.775T + 2T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 8.92T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 0.945T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 0.212T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + 1.72T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.35T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 - 1.75T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + 1.57T + 97T^{2} \) |
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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
−8.129494722336470292693174619019, −7.70647187893765997572529169470, −6.42519432996386884487434608558, −5.65087844447005309206763137180, −5.39041590623464797085448158735, −4.34856057220385692553701612814, −3.39749926116390900514272247598, −2.64448816550699213799057172250, −1.77989280716960204937371048364, 0,
1.77989280716960204937371048364, 2.64448816550699213799057172250, 3.39749926116390900514272247598, 4.34856057220385692553701612814, 5.39041590623464797085448158735, 5.65087844447005309206763137180, 6.42519432996386884487434608558, 7.70647187893765997572529169470, 8.129494722336470292693174619019