L(s) = 1 | − 2.49·5-s + 4.08·7-s − 3·9-s − 6.35·11-s − 3.31·13-s − 0.507·17-s + 19-s + 3.01·23-s + 1.23·25-s + 4.18·29-s − 9.86·31-s − 10.1·35-s − 9.92·37-s + 0.808·41-s − 5.39·43-s + 7.49·45-s + 10.4·47-s + 9.68·49-s + 10.9·53-s + 15.8·55-s + 3.91·59-s − 8.02·61-s − 12.2·63-s + 8.27·65-s + 5.98·67-s + 10.8·71-s + 0.522·73-s + ⋯ |
L(s) = 1 | − 1.11·5-s + 1.54·7-s − 9-s − 1.91·11-s − 0.919·13-s − 0.123·17-s + 0.229·19-s + 0.628·23-s + 0.246·25-s + 0.776·29-s − 1.77·31-s − 1.72·35-s − 1.63·37-s + 0.126·41-s − 0.823·43-s + 1.11·45-s + 1.53·47-s + 1.38·49-s + 1.50·53-s + 2.13·55-s + 0.509·59-s − 1.02·61-s − 1.54·63-s + 1.02·65-s + 0.731·67-s + 1.29·71-s + 0.0612·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8749237154\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8749237154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 + 0.507T + 17T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 9.86T + 31T^{2} \) |
| 37 | \( 1 + 9.92T + 37T^{2} \) |
| 41 | \( 1 - 0.808T + 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 3.91T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 0.522T + 73T^{2} \) |
| 83 | \( 1 - 2.67T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 + 7.68T + 97T^{2} \) |
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
−8.161208617629890003905586506524, −7.46402643251315561912691883138, −7.05669079779273034999048185253, −5.43704184899738425912405816356, −5.32857557375624511698157757428, −4.61231508184201739135272509323, −3.63722530991498157911466387694, −2.72522191933860605380199858900, −2.01934882041315541065709682890, −0.46562500952041516561778198642,
0.46562500952041516561778198642, 2.01934882041315541065709682890, 2.72522191933860605380199858900, 3.63722530991498157911466387694, 4.61231508184201739135272509323, 5.32857557375624511698157757428, 5.43704184899738425912405816356, 7.05669079779273034999048185253, 7.46402643251315561912691883138, 8.161208617629890003905586506524