L(s) = 1 | − 3.26·3-s − 4.10·5-s + 1.61·7-s + 7.63·9-s − 5.52·11-s + 3.26·13-s + 13.3·15-s − 4.72·17-s + 3.75·19-s − 5.25·21-s − 4.65·23-s + 11.8·25-s − 15.1·27-s − 3.54·29-s − 5.76·31-s + 18.0·33-s − 6.61·35-s − 0.102·37-s − 10.6·39-s + 10.2·41-s − 2.60·43-s − 31.3·45-s − 6.85·47-s − 4.40·49-s + 15.4·51-s + 3.77·53-s + 22.6·55-s + ⋯ |
L(s) = 1 | − 1.88·3-s − 1.83·5-s + 0.609·7-s + 2.54·9-s − 1.66·11-s + 0.904·13-s + 3.45·15-s − 1.14·17-s + 0.861·19-s − 1.14·21-s − 0.971·23-s + 2.36·25-s − 2.91·27-s − 0.658·29-s − 1.03·31-s + 3.13·33-s − 1.11·35-s − 0.0168·37-s − 1.70·39-s + 1.59·41-s − 0.396·43-s − 4.67·45-s − 1.00·47-s − 0.628·49-s + 2.15·51-s + 0.518·53-s + 3.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1599508187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1599508187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 + 0.102T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 2.60T + 43T^{2} \) |
| 47 | \( 1 + 6.85T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 + 9.91T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 6.36T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−8.114353582245910799410700786988, −7.57164860533146066027806174630, −7.14651297425982487863335563043, −6.06191933378700610959762023530, −5.48139447274228151421822538722, −4.56459904924428354139473798669, −4.31659522155290157631956338049, −3.19337412415024042613079694065, −1.56618666361384379902507205548, −0.25761067189979796360831373952,
0.25761067189979796360831373952, 1.56618666361384379902507205548, 3.19337412415024042613079694065, 4.31659522155290157631956338049, 4.56459904924428354139473798669, 5.48139447274228151421822538722, 6.06191933378700610959762023530, 7.14651297425982487863335563043, 7.57164860533146066027806174630, 8.114353582245910799410700786988