| L(s) = 1 | + 5-s − 3·7-s + 3·11-s + 6·13-s − 3·17-s − 19-s + 4·23-s − 4·25-s − 10·29-s − 2·31-s − 3·35-s − 8·37-s + 8·41-s − 43-s + 3·47-s + 2·49-s − 6·53-s + 3·55-s − 7·61-s + 6·65-s + 8·67-s + 12·71-s − 11·73-s − 9·77-s − 4·83-s − 3·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.904·11-s + 1.66·13-s − 0.727·17-s − 0.229·19-s + 0.834·23-s − 4/5·25-s − 1.85·29-s − 0.359·31-s − 0.507·35-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 0.437·47-s + 2/7·49-s − 0.824·53-s + 0.404·55-s − 0.896·61-s + 0.744·65-s + 0.977·67-s + 1.42·71-s − 1.28·73-s − 1.02·77-s − 0.439·83-s − 0.325·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.81887350203953, −16.18180407078120, −15.69769146875137, −15.22877781182094, −14.44533350536324, −13.81724510311056, −13.33926301942935, −12.89769677467479, −12.36199706404542, −11.40510641695561, −11.05769589983429, −10.47064103034283, −9.491841424759626, −9.297791790124632, −8.748240991447002, −7.937722605290241, −6.964972819076286, −6.611738286011658, −5.914911053835027, −5.504292221833288, −4.257775732469895, −3.738231292327022, −3.139452097998629, −2.035993201395619, −1.279511364527950, 0,
1.279511364527950, 2.035993201395619, 3.139452097998629, 3.738231292327022, 4.257775732469895, 5.504292221833288, 5.914911053835027, 6.611738286011658, 6.964972819076286, 7.937722605290241, 8.748240991447002, 9.297791790124632, 9.491841424759626, 10.47064103034283, 11.05769589983429, 11.40510641695561, 12.36199706404542, 12.89769677467479, 13.33926301942935, 13.81724510311056, 14.44533350536324, 15.22877781182094, 15.69769146875137, 16.18180407078120, 16.81887350203953
