L(s) = 1 | − 3-s + 7-s + 9-s + 11-s + 13-s + 3·17-s + 6·19-s − 21-s − 5·23-s − 27-s + 4·31-s − 33-s − 5·37-s − 39-s + 7·41-s + 6·43-s + 8·47-s − 6·49-s − 3·51-s + 3·53-s − 6·57-s − 12·59-s − 15·61-s + 63-s − 8·67-s + 5·69-s − 15·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.727·17-s + 1.37·19-s − 0.218·21-s − 1.04·23-s − 0.192·27-s + 0.718·31-s − 0.174·33-s − 0.821·37-s − 0.160·39-s + 1.09·41-s + 0.914·43-s + 1.16·47-s − 6/7·49-s − 0.420·51-s + 0.412·53-s − 0.794·57-s − 1.56·59-s − 1.92·61-s + 0.125·63-s − 0.977·67-s + 0.601·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−15.46682909611344, −14.77933446890895, −14.22951971403570, −13.80271370796665, −13.36406217979913, −12.48023812007994, −12.01432228088977, −11.85810398286420, −11.06701617364638, −10.58237370560034, −10.08149079180686, −9.417652849115215, −8.991231516366026, −8.176318376299157, −7.570828967337672, −7.296416247974512, −6.371166158429527, −5.801276717383914, −5.511816025209637, −4.544836735973287, −4.226283597071121, −3.297822805456646, −2.717349704528409, −1.556103779576772, −1.156956237741039, 0,
1.156956237741039, 1.556103779576772, 2.717349704528409, 3.297822805456646, 4.226283597071121, 4.544836735973287, 5.511816025209637, 5.801276717383914, 6.371166158429527, 7.296416247974512, 7.570828967337672, 8.176318376299157, 8.991231516366026, 9.417652849115215, 10.08149079180686, 10.58237370560034, 11.06701617364638, 11.85810398286420, 12.01432228088977, 12.48023812007994, 13.36406217979913, 13.80271370796665, 14.22951971403570, 14.77933446890895, 15.46682909611344