L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 5·7-s − 8-s + 9-s + 11-s − 12-s − 5·14-s + 16-s + 3·17-s − 18-s + 8·19-s − 5·21-s − 22-s + 8·23-s + 24-s − 27-s + 5·28-s − 5·29-s + 3·31-s − 32-s − 33-s − 3·34-s + 36-s − 37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.33·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.83·19-s − 1.09·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 0.192·27-s + 0.944·28-s − 0.928·29-s + 0.538·31-s − 0.176·32-s − 0.174·33-s − 0.514·34-s + 1/6·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.864957802\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864957802\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.998382162748636148134268084425, −7.50316952386372889729309031175, −7.05732818859015240038196337057, −5.87850777894753050125432631555, −5.28102076937611389130840909519, −4.75404467919835449445633335775, −3.68432362351815045007660751139, −2.60241797944573250594760075538, −1.40554548372317601550702066160, −1.00485526931738744395311643804,
1.00485526931738744395311643804, 1.40554548372317601550702066160, 2.60241797944573250594760075538, 3.68432362351815045007660751139, 4.75404467919835449445633335775, 5.28102076937611389130840909519, 5.87850777894753050125432631555, 7.05732818859015240038196337057, 7.50316952386372889729309031175, 7.998382162748636148134268084425