| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 5·11-s + 13-s − 15-s − 6·17-s + 19-s − 21-s + 23-s − 4·25-s + 5·27-s − 6·29-s − 5·33-s + 35-s + 6·37-s − 39-s − 6·41-s + 10·43-s − 2·45-s + 49-s + 6·51-s − 8·53-s + 5·55-s − 57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.229·19-s − 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s − 1.11·29-s − 0.870·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 1.52·43-s − 0.298·45-s + 1/7·49-s + 0.840·51-s − 1.09·53-s + 0.674·55-s − 0.132·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−16.38864185606972, −15.53984342123369, −15.11391396359129, −14.46674952854574, −13.89005532005359, −13.63118551497079, −12.75179599206801, −12.25749552764525, −11.51060074684870, −11.20059147284599, −10.88656852643148, −9.910913078546168, −9.204340903671106, −9.045502848383783, −8.218650728887229, −7.542247853718749, −6.623673616760216, −6.361401055142240, −5.713390204692741, −5.070942066779596, −4.267996581711652, −3.757612585544214, −2.720696879205025, −1.917870311905003, −1.149876008431362, 0,
1.149876008431362, 1.917870311905003, 2.720696879205025, 3.757612585544214, 4.267996581711652, 5.070942066779596, 5.713390204692741, 6.361401055142240, 6.623673616760216, 7.542247853718749, 8.218650728887229, 9.045502848383783, 9.204340903671106, 9.910913078546168, 10.88656852643148, 11.20059147284599, 11.51060074684870, 12.25749552764525, 12.75179599206801, 13.63118551497079, 13.89005532005359, 14.46674952854574, 15.11391396359129, 15.53984342123369, 16.38864185606972
