L(s) = 1 | + 2·5-s − 3·7-s − 6·11-s + 3·13-s + 2·17-s − 3·19-s + 6·23-s − 25-s − 8·29-s − 6·35-s − 7·37-s − 8·41-s − 12·43-s + 6·47-s + 2·49-s + 4·53-s − 12·55-s − 6·59-s + 61-s + 6·65-s − 3·67-s + 12·71-s − 15·73-s + 18·77-s − 9·79-s + 12·83-s + 4·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.13·7-s − 1.80·11-s + 0.832·13-s + 0.485·17-s − 0.688·19-s + 1.25·23-s − 1/5·25-s − 1.48·29-s − 1.01·35-s − 1.15·37-s − 1.24·41-s − 1.82·43-s + 0.875·47-s + 2/7·49-s + 0.549·53-s − 1.61·55-s − 0.781·59-s + 0.128·61-s + 0.744·65-s − 0.366·67-s + 1.42·71-s − 1.75·73-s + 2.05·77-s − 1.01·79-s + 1.31·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.003152340707084154197169692141, −8.224131195573790842679615339585, −7.24143476508618264591640621751, −6.45918568156737461680179191076, −5.63116604718208874172193979497, −5.10278218324330686292022431163, −3.62018299831604404971780816203, −2.87665746571365337045650868782, −1.78875929979941638272154943927, 0,
1.78875929979941638272154943927, 2.87665746571365337045650868782, 3.62018299831604404971780816203, 5.10278218324330686292022431163, 5.63116604718208874172193979497, 6.45918568156737461680179191076, 7.24143476508618264591640621751, 8.224131195573790842679615339585, 9.003152340707084154197169692141