L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 5·13-s − 14-s + 16-s − 3·17-s − 19-s − 6·23-s − 5·26-s − 28-s + 9·29-s − 4·31-s + 32-s − 3·34-s + 10·37-s − 38-s + 6·41-s + 10·43-s − 6·46-s − 9·47-s + 49-s − 5·52-s − 3·53-s − 56-s + 9·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s − 1.25·23-s − 0.980·26-s − 0.188·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 1.64·37-s − 0.162·38-s + 0.937·41-s + 1.52·43-s − 0.884·46-s − 1.31·47-s + 1/7·49-s − 0.693·52-s − 0.412·53-s − 0.133·56-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.531894150\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.531894150\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.63889594565132609146926416718, −6.88017730983066263649751661165, −6.27960323437322809539651535150, −5.70230531106182635796595875463, −4.71374967378600240589652395935, −4.40620659395892940657113626445, −3.48681414678244412743643711200, −2.55127517674040809756443162002, −2.12880129417094828926020137964, −0.65098955359370327824542948712,
0.65098955359370327824542948712, 2.12880129417094828926020137964, 2.55127517674040809756443162002, 3.48681414678244412743643711200, 4.40620659395892940657113626445, 4.71374967378600240589652395935, 5.70230531106182635796595875463, 6.27960323437322809539651535150, 6.88017730983066263649751661165, 7.63889594565132609146926416718