L(s) = 1 | + 3-s + 7-s − 2·9-s − 6·11-s − 5·13-s + 3·17-s + 19-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 9·29-s + 4·31-s − 6·33-s − 2·37-s − 5·39-s + 8·43-s − 6·49-s + 3·51-s + 3·53-s + 57-s + 9·59-s + 10·61-s − 2·63-s + 5·67-s − 3·69-s + 6·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.80·11-s − 1.38·13-s + 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s + 0.718·31-s − 1.04·33-s − 0.328·37-s − 0.800·39-s + 1.21·43-s − 6/7·49-s + 0.420·51-s + 0.412·53-s + 0.132·57-s + 1.17·59-s + 1.28·61-s − 0.251·63-s + 0.610·67-s − 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.488910467190776994037417053794, −8.165660732182309128185574185775, −7.944498198632498175704416365123, −7.14449686564876148233582580493, −5.56559135650201084358170066267, −5.32600053322851104543262796765, −3.98931312476793193392332292270, −2.80468657266648836263703727877, −2.16004950277651645615210004057, 0,
2.16004950277651645615210004057, 2.80468657266648836263703727877, 3.98931312476793193392332292270, 5.32600053322851104543262796765, 5.56559135650201084358170066267, 7.14449686564876148233582580493, 7.944498198632498175704416365123, 8.165660732182309128185574185775, 9.488910467190776994037417053794