| L(s) = 1 | + 2·7-s + 6·13-s + 6·17-s + 2·19-s + 8·23-s − 5·25-s − 2·29-s − 4·31-s + 2·37-s + 10·41-s + 6·43-s − 4·47-s − 3·49-s + 4·53-s + 4·59-s + 2·61-s − 8·67-s + 12·71-s + 2·73-s − 14·79-s + 4·83-s + 12·91-s + 2·97-s + 14·101-s − 16·103-s − 4·107-s − 10·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.66·13-s + 1.45·17-s + 0.458·19-s + 1.66·23-s − 25-s − 0.371·29-s − 0.718·31-s + 0.328·37-s + 1.56·41-s + 0.914·43-s − 0.583·47-s − 3/7·49-s + 0.549·53-s + 0.520·59-s + 0.256·61-s − 0.977·67-s + 1.42·71-s + 0.234·73-s − 1.57·79-s + 0.439·83-s + 1.25·91-s + 0.203·97-s + 1.39·101-s − 1.57·103-s − 0.386·107-s − 0.957·109-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
| \(L(1)\) |
\(\approx\) |
\(2.967608728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967608728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.78481499645153133392088992646, −7.24030609830940793944482121782, −6.28834417996855307354227565435, −5.62355801902291658343753559673, −5.16550895192441380031829168532, −4.11038331369023717568853462742, −3.56531018713308167210249020334, −2.70735940297540436001663397365, −1.51387314382340719639235922322, −0.954631102791004330227330112527,
0.954631102791004330227330112527, 1.51387314382340719639235922322, 2.70735940297540436001663397365, 3.56531018713308167210249020334, 4.11038331369023717568853462742, 5.16550895192441380031829168532, 5.62355801902291658343753559673, 6.28834417996855307354227565435, 7.24030609830940793944482121782, 7.78481499645153133392088992646
