| L(s) = 1 | + 2·3-s + 9-s + 3·11-s + 4·13-s − 2·19-s − 3·23-s − 4·27-s + 9·29-s − 8·31-s + 6·33-s + 5·37-s + 8·39-s + 6·41-s + 11·43-s − 6·47-s + 6·53-s − 4·57-s + 10·61-s + 5·67-s − 6·69-s + 15·71-s + 10·73-s − 7·79-s − 11·81-s − 12·83-s + 18·87-s + 12·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.904·11-s + 1.10·13-s − 0.458·19-s − 0.625·23-s − 0.769·27-s + 1.67·29-s − 1.43·31-s + 1.04·33-s + 0.821·37-s + 1.28·39-s + 0.937·41-s + 1.67·43-s − 0.875·47-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.610·67-s − 0.722·69-s + 1.78·71-s + 1.17·73-s − 0.787·79-s − 1.22·81-s − 1.31·83-s + 1.92·87-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.315751859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.315751859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.302892491342254910347877193939, −7.80026429092781153881633790359, −6.82104727985831383123636504801, −6.21179142960385677297493895179, −5.40673316579866536330795337231, −4.08308162740428403937139621322, −3.87618020070886168399849825036, −2.84294326033506919310791275790, −2.06563936265242068894460467465, −0.987839877051285196558457230133,
0.987839877051285196558457230133, 2.06563936265242068894460467465, 2.84294326033506919310791275790, 3.87618020070886168399849825036, 4.08308162740428403937139621322, 5.40673316579866536330795337231, 6.21179142960385677297493895179, 6.82104727985831383123636504801, 7.80026429092781153881633790359, 8.302892491342254910347877193939
