| L(s) = 1 | + 3-s + 7-s + 9-s + 5·13-s − 7·19-s + 21-s + 8·23-s + 27-s + 3·29-s + 6·31-s + 7·37-s + 5·39-s + 8·41-s − 8·43-s + 11·47-s + 49-s + 14·53-s − 7·57-s + 59-s + 2·61-s + 63-s − 3·67-s + 8·69-s − 8·71-s + 7·73-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.38·13-s − 1.60·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.557·29-s + 1.07·31-s + 1.15·37-s + 0.800·39-s + 1.24·41-s − 1.21·43-s + 1.60·47-s + 1/7·49-s + 1.92·53-s − 0.927·57-s + 0.130·59-s + 0.256·61-s + 0.125·63-s − 0.366·67-s + 0.963·69-s − 0.949·71-s + 0.819·73-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.472290502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.472290502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.97538664650813, −12.56422959917996, −11.77262301956766, −11.50696552403343, −10.87637677873338, −10.57184031516565, −10.19569645637145, −9.465753337982581, −8.964730716694624, −8.525466368470382, −8.439248891087688, −7.713480018169406, −7.214882223777953, −6.684429664011224, −6.187710052325246, −5.811686718160902, −5.031025776586859, −4.541007052722818, −4.055355410969809, −3.649586591879026, −2.809067444560305, −2.541127368143378, −1.824050823272646, −1.048583427334977, −0.7206332663578752,
0.7206332663578752, 1.048583427334977, 1.824050823272646, 2.541127368143378, 2.809067444560305, 3.649586591879026, 4.055355410969809, 4.541007052722818, 5.031025776586859, 5.811686718160902, 6.187710052325246, 6.684429664011224, 7.214882223777953, 7.713480018169406, 8.439248891087688, 8.525466368470382, 8.964730716694624, 9.465753337982581, 10.19569645637145, 10.57184031516565, 10.87637677873338, 11.50696552403343, 11.77262301956766, 12.56422959917996, 12.97538664650813
