L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 4·13-s + 2·15-s + 17-s − 8·19-s + 4·21-s − 25-s − 27-s − 10·31-s + 8·35-s + 8·37-s − 4·39-s + 10·41-s − 8·43-s − 2·45-s − 10·47-s + 9·49-s − 51-s − 12·53-s + 8·57-s + 8·59-s + 2·61-s − 4·63-s − 8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s + 0.242·17-s − 1.83·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s − 1.79·31-s + 1.35·35-s + 1.31·37-s − 0.640·39-s + 1.56·41-s − 1.21·43-s − 0.298·45-s − 1.45·47-s + 9/7·49-s − 0.140·51-s − 1.64·53-s + 1.05·57-s + 1.04·59-s + 0.256·61-s − 0.503·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.42179344173470, −13.44006795267049, −13.18791948540684, −12.69736115761786, −12.51179260555100, −11.77853930369687, −11.17906120805305, −10.98444646567259, −10.45605677321255, −9.702196252557447, −9.485775385186436, −8.755272422565267, −8.279881647324423, −7.727939831633495, −7.161190189439461, −6.423105309895780, −6.332070853011390, −5.784608454254453, −5.027525022465155, −4.271413119057546, −3.834844487625341, −3.487094582641426, −2.748354568308444, −1.933154907981135, −1.088670366823819, 0, 0,
1.088670366823819, 1.933154907981135, 2.748354568308444, 3.487094582641426, 3.834844487625341, 4.271413119057546, 5.027525022465155, 5.784608454254453, 6.332070853011390, 6.423105309895780, 7.161190189439461, 7.727939831633495, 8.279881647324423, 8.755272422565267, 9.485775385186436, 9.702196252557447, 10.45605677321255, 10.98444646567259, 11.17906120805305, 11.77853930369687, 12.51179260555100, 12.69736115761786, 13.18791948540684, 13.44006795267049, 14.42179344173470