L(s) = 1 | + 5-s + 0.476·7-s + 1.67·11-s + 2.67·13-s − 1.67·17-s + 7.92·19-s − 23-s + 25-s + 2.20·29-s + 6.77·31-s + 0.476·35-s + 4·37-s − 5.97·41-s − 0.402·43-s − 1.79·47-s − 6.77·49-s + 10.8·53-s + 1.67·55-s − 9.45·59-s + 6.32·61-s + 2.67·65-s + 14.7·67-s − 9.97·71-s − 6.10·73-s + 0.798·77-s − 4.49·83-s − 1.67·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.179·7-s + 0.505·11-s + 0.742·13-s − 0.406·17-s + 1.81·19-s − 0.208·23-s + 0.200·25-s + 0.408·29-s + 1.21·31-s + 0.0804·35-s + 0.657·37-s − 0.933·41-s − 0.0614·43-s − 0.262·47-s − 0.967·49-s + 1.48·53-s + 0.226·55-s − 1.23·59-s + 0.809·61-s + 0.332·65-s + 1.79·67-s − 1.18·71-s − 0.714·73-s + 0.0910·77-s − 0.493·83-s − 0.181·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.776476160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.776476160\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 0.476T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 5.97T + 41T^{2} \) |
| 43 | \( 1 + 0.402T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4.49T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
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
−7.86537482083663431555632924545, −7.01915852237603938630313080815, −6.44471021288025615794610154130, −5.75454344262708071395036682699, −5.06165794925888952068682252568, −4.30340494636804541825034726831, −3.43787744658918511573437266665, −2.72035460798698210796465671434, −1.63590201577779577115414381602, −0.884356630413398266102078288759,
0.884356630413398266102078288759, 1.63590201577779577115414381602, 2.72035460798698210796465671434, 3.43787744658918511573437266665, 4.30340494636804541825034726831, 5.06165794925888952068682252568, 5.75454344262708071395036682699, 6.44471021288025615794610154130, 7.01915852237603938630313080815, 7.86537482083663431555632924545