| L(s) = 1 | + 8·3-s + 12·5-s + 32·7-s + 37·9-s − 8·11-s − 20·13-s + 96·15-s − 98·17-s − 88·19-s + 256·21-s − 32·23-s + 19·25-s + 80·27-s + 172·29-s − 256·31-s − 64·33-s + 384·35-s + 92·37-s − 160·39-s + 102·41-s − 296·43-s + 444·45-s − 320·47-s + 681·49-s − 784·51-s + 76·53-s − 96·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 1.07·5-s + 1.72·7-s + 1.37·9-s − 0.219·11-s − 0.426·13-s + 1.65·15-s − 1.39·17-s − 1.06·19-s + 2.66·21-s − 0.290·23-s + 0.151·25-s + 0.570·27-s + 1.10·29-s − 1.48·31-s − 0.337·33-s + 1.85·35-s + 0.408·37-s − 0.656·39-s + 0.388·41-s − 1.04·43-s + 1.47·45-s − 0.993·47-s + 1.98·49-s − 2.15·51-s + 0.196·53-s − 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.898471901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.898471901\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 98 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 32 T + p^{3} T^{2} \) |
| 29 | \( 1 - 172 T + p^{3} T^{2} \) |
| 31 | \( 1 + 256 T + p^{3} T^{2} \) |
| 37 | \( 1 - 92 T + p^{3} T^{2} \) |
| 41 | \( 1 - 102 T + p^{3} T^{2} \) |
| 43 | \( 1 + 296 T + p^{3} T^{2} \) |
| 47 | \( 1 + 320 T + p^{3} T^{2} \) |
| 53 | \( 1 - 76 T + p^{3} T^{2} \) |
| 59 | \( 1 - 408 T + p^{3} T^{2} \) |
| 61 | \( 1 - 636 T + p^{3} T^{2} \) |
| 67 | \( 1 - 552 T + p^{3} T^{2} \) |
| 71 | \( 1 - 416 T + p^{3} T^{2} \) |
| 73 | \( 1 - 138 T + p^{3} T^{2} \) |
| 79 | \( 1 + 64 T + p^{3} T^{2} \) |
| 83 | \( 1 - 392 T + p^{3} T^{2} \) |
| 89 | \( 1 + 582 T + p^{3} T^{2} \) |
| 97 | \( 1 - 238 T + p^{3} T^{2} \) |
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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
−11.43813863318679460976511397426, −10.47341484486835798534393145344, −9.449641851887962456291549803023, −8.565505836094092209069868355261, −8.018379202045472140510798287986, −6.76611235353198451614794651443, −5.21397839773908649797150810808, −4.15484355161704716654809875535, −2.35826901671674391659406813542, −1.85719394803018343364894528592,
1.85719394803018343364894528592, 2.35826901671674391659406813542, 4.15484355161704716654809875535, 5.21397839773908649797150810808, 6.76611235353198451614794651443, 8.018379202045472140510798287986, 8.565505836094092209069868355261, 9.449641851887962456291549803023, 10.47341484486835798534393145344, 11.43813863318679460976511397426
