| L(s) = 1 | − 2·4-s − 5-s − 7-s − 6·11-s − 13-s + 4·16-s + 2·19-s + 2·20-s + 3·23-s + 25-s + 2·28-s − 5·31-s + 35-s − 5·37-s + 9·41-s + 8·43-s + 12·44-s + 9·47-s + 49-s + 2·52-s + 6·55-s − 5·61-s − 8·64-s + 65-s + 8·67-s + 6·71-s + 16·73-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s + 16-s + 0.458·19-s + 0.447·20-s + 0.625·23-s + 1/5·25-s + 0.377·28-s − 0.898·31-s + 0.169·35-s − 0.821·37-s + 1.40·41-s + 1.21·43-s + 1.80·44-s + 1.31·47-s + 1/7·49-s + 0.277·52-s + 0.809·55-s − 0.640·61-s − 64-s + 0.124·65-s + 0.977·67-s + 0.712·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9759327034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9759327034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.91281064690588, −13.34146818165828, −12.73578554023834, −12.57818434923842, −12.17346550856483, −11.18289621920514, −10.85494787623437, −10.45884571245122, −9.736683895932419, −9.453346983839764, −8.850471103274615, −8.324727911738908, −7.817793330764543, −7.366049974289415, −6.970614361874665, −5.860162147843884, −5.634503006518841, −5.011237848018416, −4.590065195793685, −3.877384619119351, −3.355444217388547, −2.725317011003103, −2.144705763815129, −0.9433544825599320, −0.3990723481700968,
0.3990723481700968, 0.9433544825599320, 2.144705763815129, 2.725317011003103, 3.355444217388547, 3.877384619119351, 4.590065195793685, 5.011237848018416, 5.634503006518841, 5.860162147843884, 6.970614361874665, 7.366049974289415, 7.817793330764543, 8.324727911738908, 8.850471103274615, 9.453346983839764, 9.736683895932419, 10.45884571245122, 10.85494787623437, 11.18289621920514, 12.17346550856483, 12.57818434923842, 12.73578554023834, 13.34146818165828, 13.91281064690588
