L(s) = 1 | − 7-s − 2·13-s + 19-s − 4·23-s − 5·25-s + 8·31-s + 6·37-s + 6·41-s − 4·43-s + 2·47-s + 49-s + 12·53-s + 12·59-s − 10·61-s + 4·67-s − 10·71-s − 10·73-s + 4·79-s − 14·83-s − 10·89-s + 2·91-s − 2·97-s − 8·101-s − 16·103-s + 2·107-s − 6·109-s + 20·113-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.554·13-s + 0.229·19-s − 0.834·23-s − 25-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.291·47-s + 1/7·49-s + 1.64·53-s + 1.56·59-s − 1.28·61-s + 0.488·67-s − 1.18·71-s − 1.17·73-s + 0.450·79-s − 1.53·83-s − 1.05·89-s + 0.209·91-s − 0.203·97-s − 0.796·101-s − 1.57·103-s + 0.193·107-s − 0.574·109-s + 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−7.38752565940487032749192000816, −6.66758983258427938582172242354, −5.94811082292253991209870198329, −5.42928248726281908170681118445, −4.39962640888521299992565574013, −3.97444326413970707078256062624, −2.89520014614819417543940748936, −2.32637502832526041971869622874, −1.17640750249545774676733244736, 0,
1.17640750249545774676733244736, 2.32637502832526041971869622874, 2.89520014614819417543940748936, 3.97444326413970707078256062624, 4.39962640888521299992565574013, 5.42928248726281908170681118445, 5.94811082292253991209870198329, 6.66758983258427938582172242354, 7.38752565940487032749192000816