L(s) = 1 | − 1.90·5-s + 2.81·7-s − 0.318·11-s + 5.12·13-s − 3.73·17-s − 0.725·19-s − 0.612·23-s − 1.36·25-s + 3.87·29-s − 6.65·31-s − 5.36·35-s − 9.04·37-s − 9.55·41-s − 10.9·43-s − 8.76·47-s + 0.915·49-s + 2.46·53-s + 0.608·55-s + 11.0·59-s + 12.5·61-s − 9.77·65-s + 5.21·67-s + 11.3·71-s + 2.88·73-s − 0.897·77-s + 8.98·79-s − 4.04·83-s + ⋯ |
L(s) = 1 | − 0.853·5-s + 1.06·7-s − 0.0961·11-s + 1.42·13-s − 0.905·17-s − 0.166·19-s − 0.127·23-s − 0.272·25-s + 0.720·29-s − 1.19·31-s − 0.907·35-s − 1.48·37-s − 1.49·41-s − 1.66·43-s − 1.27·47-s + 0.130·49-s + 0.338·53-s + 0.0820·55-s + 1.44·59-s + 1.60·61-s − 1.21·65-s + 0.637·67-s + 1.34·71-s + 0.337·73-s − 0.102·77-s + 1.01·79-s − 0.444·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 + 0.318T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 0.725T + 19T^{2} \) |
| 23 | \( 1 + 0.612T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 9.55T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 2.88T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 + 9.62T + 97T^{2} \) |
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
−8.005761532579491878524628664897, −6.89398598100351642635616399492, −6.56392020087178595256828961066, −5.33280076070978531478341962632, −4.93256220581892337453414578065, −3.82213246399134204474000037897, −3.59593049187975992780562017323, −2.16427006480293518396605144915, −1.38592505325482015043604847330, 0,
1.38592505325482015043604847330, 2.16427006480293518396605144915, 3.59593049187975992780562017323, 3.82213246399134204474000037897, 4.93256220581892337453414578065, 5.33280076070978531478341962632, 6.56392020087178595256828961066, 6.89398598100351642635616399492, 8.005761532579491878524628664897