L(s) = 1 | + 2·3-s + 9-s − 5·11-s − 8·17-s − 2·19-s − 7·23-s − 4·27-s + 3·29-s − 4·31-s − 10·33-s + 37-s + 2·41-s − 3·43-s − 6·47-s − 16·51-s − 10·53-s − 4·57-s − 4·59-s − 6·61-s − 13·67-s − 14·69-s + 5·71-s − 6·73-s − 13·79-s − 11·81-s + 16·83-s + 6·87-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.50·11-s − 1.94·17-s − 0.458·19-s − 1.45·23-s − 0.769·27-s + 0.557·29-s − 0.718·31-s − 1.74·33-s + 0.164·37-s + 0.312·41-s − 0.457·43-s − 0.875·47-s − 2.24·51-s − 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 1.58·67-s − 1.68·69-s + 0.593·71-s − 0.702·73-s − 1.46·79-s − 1.22·81-s + 1.75·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3220017412\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3220017412\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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
−13.97887256961283, −13.47637252691406, −13.19382446800679, −12.75131574849977, −12.11735187291923, −11.45662249970946, −10.90671680996978, −10.55006685276689, −9.914601274393199, −9.437387137118557, −8.772911202372704, −8.553103906264655, −7.789404102855903, −7.746192927645264, −6.900148852500808, −6.237106067669335, −5.832629607862472, −4.928602788201322, −4.534536755946318, −3.916024800839035, −3.160852699294151, −2.695586266619348, −2.101451831957016, −1.700455586612192, −0.1526439312750572,
0.1526439312750572, 1.700455586612192, 2.101451831957016, 2.695586266619348, 3.160852699294151, 3.916024800839035, 4.534536755946318, 4.928602788201322, 5.832629607862472, 6.237106067669335, 6.900148852500808, 7.746192927645264, 7.789404102855903, 8.553103906264655, 8.772911202372704, 9.437387137118557, 9.914601274393199, 10.55006685276689, 10.90671680996978, 11.45662249970946, 12.11735187291923, 12.75131574849977, 13.19382446800679, 13.47637252691406, 13.97887256961283