L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 11-s − 12-s − 13-s − 2·14-s + 16-s + 6·17-s − 18-s − 2·21-s + 22-s − 3·23-s + 24-s − 5·25-s + 26-s − 27-s + 2·28-s − 6·29-s − 5·31-s − 32-s + 33-s − 6·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.436·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.898·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.256105886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.256105886\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.57921646755944, −12.07581399065949, −11.68110557156445, −11.22173043857688, −10.85634974549977, −10.40395190508803, −9.896837095985219, −9.470565728833757, −9.179783847284196, −8.311766363948851, −8.017472526485795, −7.593315187370433, −7.284556476489919, −6.637556380776465, −5.991253282728900, −5.590425811035515, −5.272313489290458, −4.680720412159921, −3.848112509029157, −3.657955201150976, −2.799147785848829, −2.033512228791913, −1.784788627248320, −0.9961967785223025, −0.3934613516791840,
0.3934613516791840, 0.9961967785223025, 1.784788627248320, 2.033512228791913, 2.799147785848829, 3.657955201150976, 3.848112509029157, 4.680720412159921, 5.272313489290458, 5.590425811035515, 5.991253282728900, 6.637556380776465, 7.284556476489919, 7.593315187370433, 8.017472526485795, 8.311766363948851, 9.179783847284196, 9.470565728833757, 9.896837095985219, 10.40395190508803, 10.85634974549977, 11.22173043857688, 11.68110557156445, 12.07581399065949, 12.57921646755944