L(s) = 1 | − 4·7-s − 4·11-s + 13-s − 8·17-s − 8·19-s + 8·23-s − 4·29-s + 31-s + 3·37-s − 11·43-s + 8·47-s + 9·49-s + 12·53-s − 8·59-s − 2·61-s + 11·67-s − 12·71-s + 9·73-s + 16·77-s + 9·79-s + 4·83-s + 12·89-s − 4·91-s − 2·97-s − 5·103-s − 12·107-s + 5·109-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.277·13-s − 1.94·17-s − 1.83·19-s + 1.66·23-s − 0.742·29-s + 0.179·31-s + 0.493·37-s − 1.67·43-s + 1.16·47-s + 9/7·49-s + 1.64·53-s − 1.04·59-s − 0.256·61-s + 1.34·67-s − 1.42·71-s + 1.05·73-s + 1.82·77-s + 1.01·79-s + 0.439·83-s + 1.27·89-s − 0.419·91-s − 0.203·97-s − 0.492·103-s − 1.16·107-s + 0.478·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6776962528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6776962528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.322260356955643509234888612937, −7.29011109579095952052581471235, −6.64211081847536612783118200688, −6.25466828737871731568861226632, −5.27640503156460691381819247605, −4.47637965948201297233639578398, −3.66179085212812445606160031322, −2.74011041907026224782874095862, −2.15645908513816520114343713801, −0.40902501861440552958568419975,
0.40902501861440552958568419975, 2.15645908513816520114343713801, 2.74011041907026224782874095862, 3.66179085212812445606160031322, 4.47637965948201297233639578398, 5.27640503156460691381819247605, 6.25466828737871731568861226632, 6.64211081847536612783118200688, 7.29011109579095952052581471235, 8.322260356955643509234888612937