| L(s) = 1 | − 5-s − 2·7-s − 2·13-s − 6·17-s + 23-s + 25-s + 2·35-s − 8·37-s + 12·41-s − 2·43-s − 8·47-s − 3·49-s + 10·53-s − 14·59-s + 2·61-s + 2·65-s + 2·67-s − 2·71-s − 6·73-s + 4·83-s + 6·85-s + 10·89-s + 4·91-s − 12·97-s + 14·103-s − 4·107-s + 18·109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.208·23-s + 1/5·25-s + 0.338·35-s − 1.31·37-s + 1.87·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 1.82·59-s + 0.256·61-s + 0.248·65-s + 0.244·67-s − 0.237·71-s − 0.702·73-s + 0.439·83-s + 0.650·85-s + 1.05·89-s + 0.419·91-s − 1.21·97-s + 1.37·103-s − 0.386·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9120276995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9120276995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72731399349194171633351700328, −7.05811930419239877686318829119, −6.54989325780027366059452350251, −5.81809669901073407894839535109, −4.88018072312577552211371275346, −4.31633442111306918700954283314, −3.46109277812552102231847707648, −2.73712448173452933626651442096, −1.84632565661201687966394322473, −0.44925806807779103234323865527,
0.44925806807779103234323865527, 1.84632565661201687966394322473, 2.73712448173452933626651442096, 3.46109277812552102231847707648, 4.31633442111306918700954283314, 4.88018072312577552211371275346, 5.81809669901073407894839535109, 6.54989325780027366059452350251, 7.05811930419239877686318829119, 7.72731399349194171633351700328
