| L(s) = 1 | − 2·5-s − 2·7-s − 6·17-s + 6·19-s + 4·23-s − 25-s + 2·29-s − 8·31-s + 4·35-s + 2·37-s − 10·41-s + 10·43-s − 3·49-s − 6·53-s − 4·61-s − 4·67-s + 4·73-s − 10·79-s − 12·83-s + 12·85-s − 18·89-s − 12·95-s + 18·97-s + 18·101-s + 8·103-s − 4·107-s + 16·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1.45·17-s + 1.37·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 1.52·43-s − 3/7·49-s − 0.824·53-s − 0.512·61-s − 0.488·67-s + 0.468·73-s − 1.12·79-s − 1.31·83-s + 1.30·85-s − 1.90·89-s − 1.23·95-s + 1.82·97-s + 1.79·101-s + 0.788·103-s − 0.386·107-s + 1.53·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9176083712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9176083712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−7.55372985967930080319019344694, −7.21526436143177693708150987215, −6.49496482978048080646157049851, −5.73307799720136701794649921720, −4.89321573633178671016398543508, −4.20816283126667218064419253358, −3.41789851877515062432525133342, −2.87604403867179592681459689738, −1.72160698518041378216572130755, −0.45715230487275096972507215382,
0.45715230487275096972507215382, 1.72160698518041378216572130755, 2.87604403867179592681459689738, 3.41789851877515062432525133342, 4.20816283126667218064419253358, 4.89321573633178671016398543508, 5.73307799720136701794649921720, 6.49496482978048080646157049851, 7.21526436143177693708150987215, 7.55372985967930080319019344694
