| 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
| \(L(1)\) |
\(\approx\) |
\(2.675710673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675710673\) |
| \(L(\frac{3}{2})\) |
|
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
−9.287440446093371828678040449410, −8.650680906201277123838913846622, −7.78639326445418573570183285345, −6.88359362105880509746395731542, −5.95444690232608248894153675717, −5.45584155430329122752354767088, −4.24643980416916525601100537552, −3.53215777188464305495336430815, −1.89419396016353367846966262393, −1.36990894878106263941737695602,
1.36990894878106263941737695602, 1.89419396016353367846966262393, 3.53215777188464305495336430815, 4.24643980416916525601100537552, 5.45584155430329122752354767088, 5.95444690232608248894153675717, 6.88359362105880509746395731542, 7.78639326445418573570183285345, 8.650680906201277123838913846622, 9.287440446093371828678040449410
