| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s + 4·13-s + 16-s + 6·17-s − 19-s + 4·22-s + 4·26-s + 2·29-s + 32-s + 6·34-s + 4·37-s − 38-s + 12·41-s − 6·43-s + 4·44-s − 7·49-s + 4·52-s − 14·53-s + 2·58-s + 10·59-s − 6·61-s + 64-s + 4·67-s + 6·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 0.852·22-s + 0.784·26-s + 0.371·29-s + 0.176·32-s + 1.02·34-s + 0.657·37-s − 0.162·38-s + 1.87·41-s − 0.914·43-s + 0.603·44-s − 49-s + 0.554·52-s − 1.92·53-s + 0.262·58-s + 1.30·59-s − 0.768·61-s + 1/8·64-s + 0.488·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.326511230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.326511230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.75206203065782824564601025977, −6.93983268612620495893893766416, −6.15436139639844471974832715314, −5.93256107176594430763754541291, −4.91759583883050951433046623960, −4.19682039163811492769679090553, −3.54164492977917525621544952031, −2.92634822754507329651123531257, −1.68729660447326886129255280761, −1.00661548576639621846125273439,
1.00661548576639621846125273439, 1.68729660447326886129255280761, 2.92634822754507329651123531257, 3.54164492977917525621544952031, 4.19682039163811492769679090553, 4.91759583883050951433046623960, 5.93256107176594430763754541291, 6.15436139639844471974832715314, 6.93983268612620495893893766416, 7.75206203065782824564601025977
