| L(s) = 1 | + 18·5-s − 7·7-s − 44·11-s + 58·13-s + 130·17-s + 92·19-s − 84·23-s + 199·25-s + 250·29-s − 72·31-s − 126·35-s − 354·37-s − 334·41-s − 416·43-s + 464·47-s + 49·49-s + 450·53-s − 792·55-s + 516·59-s + 58·61-s + 1.04e3·65-s − 656·67-s + 940·71-s + 178·73-s + 308·77-s + 1.07e3·79-s − 660·83-s + ⋯ |
| L(s) = 1 | + 1.60·5-s − 0.377·7-s − 1.20·11-s + 1.23·13-s + 1.85·17-s + 1.11·19-s − 0.761·23-s + 1.59·25-s + 1.60·29-s − 0.417·31-s − 0.608·35-s − 1.57·37-s − 1.27·41-s − 1.47·43-s + 1.44·47-s + 1/7·49-s + 1.16·53-s − 1.94·55-s + 1.13·59-s + 0.121·61-s + 1.99·65-s − 1.19·67-s + 1.57·71-s + 0.285·73-s + 0.455·77-s + 1.52·79-s − 0.872·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.352310688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.352310688\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 250 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 354 T + p^{3} T^{2} \) |
| 41 | \( 1 + 334 T + p^{3} T^{2} \) |
| 43 | \( 1 + 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 516 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 656 T + p^{3} T^{2} \) |
| 71 | \( 1 - 940 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1072 T + p^{3} T^{2} \) |
| 83 | \( 1 + 660 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1254 T + p^{3} T^{2} \) |
| 97 | \( 1 - 210 T + p^{3} T^{2} \) |
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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
−8.772769527387875393582609274072, −8.159394425372046396279031835009, −7.12676532139105332027310339333, −6.28969039910220894493135407905, −5.42273004623438692173930303367, −5.29286284757567950944370270478, −3.59299801641857320991402126637, −2.88267782853898508878979437408, −1.79577611935370511146998836438, −0.881534157530741021753192307815,
0.881534157530741021753192307815, 1.79577611935370511146998836438, 2.88267782853898508878979437408, 3.59299801641857320991402126637, 5.29286284757567950944370270478, 5.42273004623438692173930303367, 6.28969039910220894493135407905, 7.12676532139105332027310339333, 8.159394425372046396279031835009, 8.772769527387875393582609274072
