| L(s) = 1 | − 5·3-s − 2·7-s − 2·9-s − 39·11-s + 84·13-s − 61·17-s − 151·19-s + 10·21-s + 58·23-s + 145·27-s + 192·29-s + 18·31-s + 195·33-s − 138·37-s − 420·39-s + 229·41-s + 164·43-s + 212·47-s − 339·49-s + 305·51-s + 578·53-s + 755·57-s + 336·59-s + 858·61-s + 4·63-s + 209·67-s − 290·69-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.107·7-s − 0.0740·9-s − 1.06·11-s + 1.79·13-s − 0.870·17-s − 1.82·19-s + 0.103·21-s + 0.525·23-s + 1.03·27-s + 1.22·29-s + 0.104·31-s + 1.02·33-s − 0.613·37-s − 1.72·39-s + 0.872·41-s + 0.581·43-s + 0.657·47-s − 0.988·49-s + 0.837·51-s + 1.49·53-s + 1.75·57-s + 0.741·59-s + 1.80·61-s + 0.00799·63-s + 0.381·67-s − 0.505·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.008468385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008468385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 - 84 T + p^{3} T^{2} \) |
| 17 | \( 1 + 61 T + p^{3} T^{2} \) |
| 19 | \( 1 + 151 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 192 T + p^{3} T^{2} \) |
| 31 | \( 1 - 18 T + p^{3} T^{2} \) |
| 37 | \( 1 + 138 T + p^{3} T^{2} \) |
| 41 | \( 1 - 229 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 212 T + p^{3} T^{2} \) |
| 53 | \( 1 - 578 T + p^{3} T^{2} \) |
| 59 | \( 1 - 336 T + p^{3} T^{2} \) |
| 61 | \( 1 - 858 T + p^{3} T^{2} \) |
| 67 | \( 1 - 209 T + p^{3} T^{2} \) |
| 71 | \( 1 - 780 T + p^{3} T^{2} \) |
| 73 | \( 1 + 403 T + p^{3} T^{2} \) |
| 79 | \( 1 - 230 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1293 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1369 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−10.83012286836527192786361856257, −10.40070081007949510094835118567, −8.799856491221432216672537302209, −8.295762985863758344977004589960, −6.74139499385962362944463158292, −6.12506860156555018369553053888, −5.14539448262131903101312202440, −4.00597242364009796389549547487, −2.46491234231167501695306917229, −0.67265161510883313977865450518,
0.67265161510883313977865450518, 2.46491234231167501695306917229, 4.00597242364009796389549547487, 5.14539448262131903101312202440, 6.12506860156555018369553053888, 6.74139499385962362944463158292, 8.295762985863758344977004589960, 8.799856491221432216672537302209, 10.40070081007949510094835118567, 10.83012286836527192786361856257
