L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 6·11-s + 13-s + 4·14-s + 16-s + 4·17-s + 2·19-s − 6·22-s + 6·23-s − 26-s − 4·28-s + 10·29-s + 4·31-s − 32-s − 4·34-s − 6·37-s − 2·38-s − 10·41-s + 6·44-s − 6·46-s − 8·47-s + 9·49-s + 52-s + 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 1.80·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 1.27·22-s + 1.25·23-s − 0.196·26-s − 0.755·28-s + 1.85·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.986·37-s − 0.324·38-s − 1.56·41-s + 0.904·44-s − 0.884·46-s − 1.16·47-s + 9/7·49-s + 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.431416223\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431416223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.339904376267732337500364977299, −7.19490360721045366899290328779, −6.69541915346419258645546151083, −6.34011968070389437571269259261, −5.41382098344688334785610106041, −4.32665751079480153167990537688, −3.24461773165466241188872526708, −3.10842757006101784198516456238, −1.53234195499874651444031597665, −0.75971046034338922492416735899,
0.75971046034338922492416735899, 1.53234195499874651444031597665, 3.10842757006101784198516456238, 3.24461773165466241188872526708, 4.32665751079480153167990537688, 5.41382098344688334785610106041, 6.34011968070389437571269259261, 6.69541915346419258645546151083, 7.19490360721045366899290328779, 8.339904376267732337500364977299