L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 13-s + 3·17-s − 2·19-s + 4·21-s − 6·23-s + 4·27-s + 29-s − 10·31-s − 3·37-s + 2·39-s − 11·41-s + 12·43-s + 10·47-s − 3·49-s − 6·51-s + 9·53-s + 4·57-s − 4·59-s + 6·61-s − 2·63-s + 2·67-s + 12·69-s + 8·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s + 0.185·29-s − 1.79·31-s − 0.493·37-s + 0.320·39-s − 1.71·41-s + 1.82·43-s + 1.45·47-s − 3/7·49-s − 0.840·51-s + 1.23·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s + 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.16370495170553, −12.68988528093830, −12.25190199874853, −12.11259092529864, −11.40039447690197, −11.07273022349994, −10.44514909053028, −10.15311024168557, −9.753671890262741, −9.050595692785200, −8.662728659348648, −8.080606035108965, −7.267891224023113, −7.171928119332375, −6.447564713380350, −5.958053091190957, −5.616328940710402, −5.230322041099473, −4.505238610554697, −3.916125843563024, −3.493029907718031, −2.735784393761231, −2.131039234632332, −1.392885713368019, −0.5570265396521246, 0,
0.5570265396521246, 1.392885713368019, 2.131039234632332, 2.735784393761231, 3.493029907718031, 3.916125843563024, 4.505238610554697, 5.230322041099473, 5.616328940710402, 5.958053091190957, 6.447564713380350, 7.171928119332375, 7.267891224023113, 8.080606035108965, 8.662728659348648, 9.050595692785200, 9.753671890262741, 10.15311024168557, 10.44514909053028, 11.07273022349994, 11.40039447690197, 12.11259092529864, 12.25190199874853, 12.68988528093830, 13.16370495170553