L(s) = 1 | + 5-s + 7-s − 2·11-s + 6·13-s − 2·19-s + 23-s + 25-s + 4·29-s − 10·31-s + 35-s + 10·37-s + 6·41-s − 6·43-s + 10·47-s + 49-s − 10·53-s − 2·55-s − 4·59-s + 6·61-s + 6·65-s + 2·67-s − 2·71-s − 10·73-s − 2·77-s − 16·79-s + 6·83-s + 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.66·13-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.742·29-s − 1.79·31-s + 0.169·35-s + 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.45·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s − 0.520·59-s + 0.768·61-s + 0.744·65-s + 0.244·67-s − 0.237·71-s − 1.17·73-s − 0.227·77-s − 1.80·79-s + 0.658·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−13.80610096229130, −13.26474974089104, −12.93274282769170, −12.64145626566403, −11.77022706856731, −11.35757118771769, −10.86897649933900, −10.56719222065563, −10.04765580731906, −9.244196201593625, −9.033503260138608, −8.428425156478605, −7.944262044043800, −7.471894895651591, −6.811750935911235, −6.160992733184140, −5.889643152262068, −5.311437593926974, −4.673707466535370, −4.089647380845257, −3.568832436223087, −2.816345784600445, −2.303910204213944, −1.488368111917740, −1.049025852671556, 0,
1.049025852671556, 1.488368111917740, 2.303910204213944, 2.816345784600445, 3.568832436223087, 4.089647380845257, 4.673707466535370, 5.311437593926974, 5.889643152262068, 6.160992733184140, 6.811750935911235, 7.471894895651591, 7.944262044043800, 8.428425156478605, 9.033503260138608, 9.244196201593625, 10.04765580731906, 10.56719222065563, 10.86897649933900, 11.35757118771769, 11.77022706856731, 12.64145626566403, 12.93274282769170, 13.26474974089104, 13.80610096229130