| L(s) = 1 | − 3-s − 3·7-s + 9-s + 3·11-s − 13-s + 3·17-s + 3·21-s − 3·23-s − 27-s − 8·29-s − 4·31-s − 3·33-s − 37-s + 39-s − 3·41-s − 4·43-s + 10·47-s + 2·49-s − 3·51-s + 9·53-s + 4·59-s − 9·61-s − 3·63-s + 4·67-s + 3·69-s − 7·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.654·21-s − 0.625·23-s − 0.192·27-s − 1.48·29-s − 0.718·31-s − 0.522·33-s − 0.164·37-s + 0.160·39-s − 0.468·41-s − 0.609·43-s + 1.45·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s + 0.520·59-s − 1.15·61-s − 0.377·63-s + 0.488·67-s + 0.361·69-s − 0.830·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−14.53264679510859, −14.01983304281757, −13.33085874662021, −13.08219456784149, −12.28728062280335, −12.14591781156439, −11.58565099505591, −10.99395302767968, −10.34817496348198, −10.00384884673972, −9.396542113161127, −9.073506507929871, −8.414629743881763, −7.462071950719419, −7.317798636065394, −6.606193688487286, −6.051105867753147, −5.706985226053479, −5.048966063411765, −4.276054553417942, −3.616573564813309, −3.377900810382318, −2.338037758417821, −1.683172146709841, −0.7856529643470219, 0,
0.7856529643470219, 1.683172146709841, 2.338037758417821, 3.377900810382318, 3.616573564813309, 4.276054553417942, 5.048966063411765, 5.706985226053479, 6.051105867753147, 6.606193688487286, 7.317798636065394, 7.462071950719419, 8.414629743881763, 9.073506507929871, 9.396542113161127, 10.00384884673972, 10.34817496348198, 10.99395302767968, 11.58565099505591, 12.14591781156439, 12.28728062280335, 13.08219456784149, 13.33085874662021, 14.01983304281757, 14.53264679510859
