L(s) = 1 | − 5-s + 4·7-s − 2·13-s + 6·17-s − 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s + 6·37-s + 6·41-s − 12·43-s − 12·47-s + 9·49-s + 10·53-s + 8·59-s − 10·61-s + 2·65-s + 12·67-s + 8·71-s + 10·73-s − 16·79-s + 12·83-s − 6·85-s + 6·89-s − 8·91-s + 18·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.554·13-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.82·43-s − 1.75·47-s + 9/7·49-s + 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 1.31·83-s − 0.650·85-s + 0.635·89-s − 0.838·91-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.882772274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.882772274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.688118946800696446163690263696, −8.321691453310137231597200178920, −8.104141162644996210965549984793, −7.33091456015585229923068630884, −6.22872316322800312854769818070, −5.16433158810068959087845034261, −4.60827511198304462073880960004, −3.53473199255158139928482636763, −2.28702756907575085464239244413, −1.05420551230126044240437075750,
1.05420551230126044240437075750, 2.28702756907575085464239244413, 3.53473199255158139928482636763, 4.60827511198304462073880960004, 5.16433158810068959087845034261, 6.22872316322800312854769818070, 7.33091456015585229923068630884, 8.104141162644996210965549984793, 8.321691453310137231597200178920, 9.688118946800696446163690263696