L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 11-s − 3·13-s − 14-s + 16-s + 2·17-s − 3·18-s + 5·19-s − 22-s + 7·23-s − 3·26-s − 28-s + 3·29-s − 3·31-s + 32-s + 2·34-s − 3·36-s + 8·37-s + 5·38-s − 11·43-s − 44-s + 7·46-s + 8·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.14·19-s − 0.213·22-s + 1.45·23-s − 0.588·26-s − 0.188·28-s + 0.557·29-s − 0.538·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.31·37-s + 0.811·38-s − 1.67·43-s − 0.150·44-s + 1.03·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.580355754\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.580355754\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−8.352054853371123821033284853939, −7.65558100473630831122600941631, −6.93027941342444787375688904823, −6.19869388198090898681557576358, −5.16057128352108511744525826145, −5.11183024607625390725057160754, −3.71651330528317709288293448594, −3.02510092100263003661086566599, −2.36294746398274522663361775067, −0.823263731924689320087869435652,
0.823263731924689320087869435652, 2.36294746398274522663361775067, 3.02510092100263003661086566599, 3.71651330528317709288293448594, 5.11183024607625390725057160754, 5.16057128352108511744525826145, 6.19869388198090898681557576358, 6.93027941342444787375688904823, 7.65558100473630831122600941631, 8.352054853371123821033284853939