L(s) = 1 | − 5-s − 7-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s − 23-s + 25-s − 2·29-s + 35-s + 6·37-s + 6·41-s + 8·43-s + 4·47-s + 49-s − 2·53-s + 4·55-s + 12·59-s − 2·61-s − 2·65-s − 8·67-s + 8·71-s − 2·73-s + 4·77-s − 8·79-s + 8·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.169·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 0.878·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.125124818\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.125124818\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.59139301676950, −12.99606341235292, −12.76380020963870, −12.17102204827296, −11.65364048193276, −11.15489007532000, −10.72429386272653, −10.20771339421906, −9.725067048697681, −9.194056553344360, −8.697666614980045, −8.022321865245024, −7.618523599269354, −7.362094256484874, −6.580609964359352, −5.929926708654718, −5.590622848634291, −5.016479725689010, −4.306977328215248, −3.819814230630272, −3.157536259437812, −2.701050331080393, −2.041777618622820, −1.030684326688249, −0.5260437575680942,
0.5260437575680942, 1.030684326688249, 2.041777618622820, 2.701050331080393, 3.157536259437812, 3.819814230630272, 4.306977328215248, 5.016479725689010, 5.590622848634291, 5.929926708654718, 6.580609964359352, 7.362094256484874, 7.618523599269354, 8.022321865245024, 8.697666614980045, 9.194056553344360, 9.725067048697681, 10.20771339421906, 10.72429386272653, 11.15489007532000, 11.65364048193276, 12.17102204827296, 12.76380020963870, 12.99606341235292, 13.59139301676950