L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 2·13-s + 14-s + 16-s + 4·19-s + 2·20-s − 25-s + 2·26-s − 28-s − 6·29-s − 32-s − 2·35-s + 6·37-s − 4·38-s − 2·40-s − 6·41-s − 12·43-s − 8·47-s + 49-s + 50-s − 2·52-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.338·35-s + 0.986·37-s − 0.648·38-s − 0.316·40-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.141·50-s − 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.287578095\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287578095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.01776894565797, −14.38430367973300, −13.85216630146790, −13.25435090213305, −12.93663762004685, −12.19894527318492, −11.60056490325862, −11.27259474059675, −10.43381752536904, −9.986825045332634, −9.542863611500645, −9.309548082574135, −8.444204757369049, −7.957553377478190, −7.325533325167628, −6.724887860070433, −6.284590030031548, −5.472725108529631, −5.214824857605784, −4.244783463156350, −3.368238357968954, −2.864641847387471, −1.981061820445788, −1.545745240165770, −0.4594645069333180,
0.4594645069333180, 1.545745240165770, 1.981061820445788, 2.864641847387471, 3.368238357968954, 4.244783463156350, 5.214824857605784, 5.472725108529631, 6.284590030031548, 6.724887860070433, 7.325533325167628, 7.957553377478190, 8.444204757369049, 9.309548082574135, 9.542863611500645, 9.986825045332634, 10.43381752536904, 11.27259474059675, 11.60056490325862, 12.19894527318492, 12.93663762004685, 13.25435090213305, 13.85216630146790, 14.38430367973300, 15.01776894565797