L(s) = 1 | − 5-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s − 8·43-s + 3·45-s − 4·47-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s + 2·85-s + 6·89-s + 4·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.526149348\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526149348\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.938571023394552259974771089914, −8.568492662536040706055181654407, −7.73148938409257749345471205858, −6.56407051249131463125622663292, −6.27788039497575849944468549246, −5.08516938187217910881281818728, −4.17940013953949332311038958729, −3.38301616382596410288851087260, −2.29215136886722219541328805003, −0.834108506722948485247590602076,
0.834108506722948485247590602076, 2.29215136886722219541328805003, 3.38301616382596410288851087260, 4.17940013953949332311038958729, 5.08516938187217910881281818728, 6.27788039497575849944468549246, 6.56407051249131463125622663292, 7.73148938409257749345471205858, 8.568492662536040706055181654407, 8.938571023394552259974771089914