| L(s) = 1 | + 4·7-s + 11-s − 13-s − 2·17-s + 4·19-s − 4·23-s − 6·29-s − 10·37-s + 10·41-s − 4·43-s + 9·49-s + 10·53-s + 12·59-s − 2·61-s − 8·67-s + 8·71-s + 2·73-s + 4·77-s − 16·79-s − 12·83-s + 10·89-s − 4·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s − 1.64·37-s + 1.56·41-s − 0.609·43-s + 9/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s + 0.949·71-s + 0.234·73-s + 0.455·77-s − 1.80·79-s − 1.31·83-s + 1.05·89-s − 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.05531582092961, −12.60023843254225, −11.89426594265630, −11.66853129534234, −11.37054504082071, −10.81658403384214, −10.23645869469460, −9.967389935798124, −9.232685306122691, −8.753490017718752, −8.530965666765850, −7.802825493806735, −7.399495878551196, −7.189738208863052, −6.383881303409351, −5.801433451023310, −5.377396433572393, −4.902157648723710, −4.424403947060865, −3.818946740362462, −3.429831260424610, −2.450951528492047, −2.102291099566826, −1.499990420984328, −0.9085784981482994, 0,
0.9085784981482994, 1.499990420984328, 2.102291099566826, 2.450951528492047, 3.429831260424610, 3.818946740362462, 4.424403947060865, 4.902157648723710, 5.377396433572393, 5.801433451023310, 6.383881303409351, 7.189738208863052, 7.399495878551196, 7.802825493806735, 8.530965666765850, 8.753490017718752, 9.232685306122691, 9.967389935798124, 10.23645869469460, 10.81658403384214, 11.37054504082071, 11.66853129534234, 11.89426594265630, 12.60023843254225, 13.05531582092961
