L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.981020375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981020375\) |
\(L(1)\) |
\(\approx\) |
\(1.835836955\) |
\(L(1)\) |
\(\approx\) |
\(1.835836955\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.92503322763303281849392160292, −28.30346814709051935989006213871, −26.175562985130485956339608171218, −25.76011720626284665983809601606, −24.72517076905428831589850174120, −23.58333578895241070728058294509, −22.710600015160979431809668071198, −21.717030616432443637056296311, −20.96796244708498986455341065005, −19.919152580134126358970449134840, −18.65326493193326148316260679010, −17.372559097162408186160883128543, −16.10446700917013154437614951768, −15.423827187306917393909829844, −13.86138053952253514514626317218, −13.31420313500417711586281266252, −12.41750639880045620278095023399, −10.82209265347298076555966000015, −10.03163214254246147181169030819, −8.42634781748281209173830364905, −6.62255494014020266337495914010, −6.058358901333976442032144499644, −4.69999004651383639972471567304, −3.192000650645312588512587572055, −2.03656834886337276049517659329,
2.03656834886337276049517659329, 3.192000650645312588512587572055, 4.69999004651383639972471567304, 6.058358901333976442032144499644, 6.62255494014020266337495914010, 8.42634781748281209173830364905, 10.03163214254246147181169030819, 10.82209265347298076555966000015, 12.41750639880045620278095023399, 13.31420313500417711586281266252, 13.86138053952253514514626317218, 15.423827187306917393909829844, 16.10446700917013154437614951768, 17.372559097162408186160883128543, 18.65326493193326148316260679010, 19.919152580134126358970449134840, 20.96796244708498986455341065005, 21.717030616432443637056296311, 22.710600015160979431809668071198, 23.58333578895241070728058294509, 24.72517076905428831589850174120, 25.76011720626284665983809601606, 26.175562985130485956339608171218, 28.30346814709051935989006213871, 28.92503322763303281849392160292