| L(s) = 1 | − 3-s − 7-s + 9-s − 5·11-s + 17-s + 21-s − 4·23-s − 27-s − 3·29-s − 31-s + 5·33-s − 37-s − 41-s − 7·43-s + 4·47-s − 6·49-s − 51-s + 3·53-s + 8·59-s + 5·61-s − 63-s + 4·67-s + 4·69-s + 6·71-s + 10·73-s + 5·77-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.242·17-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.557·29-s − 0.179·31-s + 0.870·33-s − 0.164·37-s − 0.156·41-s − 1.06·43-s + 0.583·47-s − 6/7·49-s − 0.140·51-s + 0.412·53-s + 1.04·59-s + 0.640·61-s − 0.125·63-s + 0.488·67-s + 0.481·69-s + 0.712·71-s + 1.17·73-s + 0.569·77-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.91763435881279, −14.51522856689338, −13.67882364279183, −13.32364520405818, −12.89873138017685, −12.30919282172280, −11.89013486871540, −11.23602637860773, −10.71215616900759, −10.30321576981172, −9.745477342438629, −9.346613558732931, −8.339789265646262, −8.114991148230200, −7.429122904511752, −6.851582876664340, −6.304267456152641, −5.537944835350960, −5.306099890242359, −4.627507632904207, −3.811344857388374, −3.288193762900090, −2.415915356215707, −1.884337375564542, −0.7496723993408093, 0,
0.7496723993408093, 1.884337375564542, 2.415915356215707, 3.288193762900090, 3.811344857388374, 4.627507632904207, 5.306099890242359, 5.537944835350960, 6.304267456152641, 6.851582876664340, 7.429122904511752, 8.114991148230200, 8.339789265646262, 9.346613558732931, 9.745477342438629, 10.30321576981172, 10.71215616900759, 11.23602637860773, 11.89013486871540, 12.30919282172280, 12.89873138017685, 13.32364520405818, 13.67882364279183, 14.51522856689338, 14.91763435881279
