L(s) = 1 | − 5-s + 2·11-s + 13-s + 6·17-s + 6·19-s − 6·23-s + 25-s + 6·29-s + 6·31-s + 6·37-s − 10·41-s − 6·43-s − 6·53-s − 2·55-s − 10·59-s − 2·61-s − 65-s + 8·67-s + 10·71-s + 14·73-s − 12·79-s + 12·83-s − 6·85-s − 10·89-s − 6·95-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.986·37-s − 1.56·41-s − 0.914·43-s − 0.824·53-s − 0.269·55-s − 1.30·59-s − 0.256·61-s − 0.124·65-s + 0.977·67-s + 1.18·71-s + 1.63·73-s − 1.35·79-s + 1.31·83-s − 0.650·85-s − 1.05·89-s − 0.615·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.25116874433575, −12.41419924904178, −12.18438958873908, −11.91749246358304, −11.36310646315736, −10.98298780191953, −10.17665583582365, −9.869787927113840, −9.652175119976272, −8.922131173525940, −8.251990027233107, −8.048986633698447, −7.637736286061833, −6.944680546147835, −6.472112091038437, −6.053578055261977, −5.437063755288063, −4.883466435231758, −4.481532702019211, −3.614958042339564, −3.464552452335473, −2.866224804422660, −2.098183488033188, −1.220261875264202, −1.013521376051489, 0,
1.013521376051489, 1.220261875264202, 2.098183488033188, 2.866224804422660, 3.464552452335473, 3.614958042339564, 4.481532702019211, 4.883466435231758, 5.437063755288063, 6.053578055261977, 6.472112091038437, 6.944680546147835, 7.637736286061833, 8.048986633698447, 8.251990027233107, 8.922131173525940, 9.652175119976272, 9.869787927113840, 10.17665583582365, 10.98298780191953, 11.36310646315736, 11.91749246358304, 12.18438958873908, 12.41419924904178, 13.25116874433575