L(s) = 1 | + 3-s + 9-s − 2·13-s − 6·17-s − 8·19-s + 27-s − 6·29-s − 4·31-s − 10·37-s − 2·39-s + 6·41-s + 4·43-s − 6·51-s − 6·53-s − 8·57-s + 12·59-s − 10·61-s + 4·67-s − 12·71-s − 10·73-s − 8·79-s + 81-s + 12·83-s − 6·87-s + 6·89-s − 4·93-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 0.635·89-s − 0.414·93-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39182472169175, −12.98075227159389, −12.47904392354552, −12.16093473023371, −11.43181763635467, −10.93028731629939, −10.65489357943359, −10.20746505442130, −9.388516738917597, −9.237244651703120, −8.658753507442678, −8.372451796469225, −7.688385225097239, −7.215404268717093, −6.816974531788808, −6.295682850669530, −5.737075948726264, −5.136348808087400, −4.480614623870300, −4.161115211791177, −3.656346806832715, −2.914317150152996, −2.288018242282653, −2.008495125104859, −1.329078496122330, 0, 0,
1.329078496122330, 2.008495125104859, 2.288018242282653, 2.914317150152996, 3.656346806832715, 4.161115211791177, 4.480614623870300, 5.136348808087400, 5.737075948726264, 6.295682850669530, 6.816974531788808, 7.215404268717093, 7.688385225097239, 8.372451796469225, 8.658753507442678, 9.237244651703120, 9.388516738917597, 10.20746505442130, 10.65489357943359, 10.93028731629939, 11.43181763635467, 12.16093473023371, 12.47904392354552, 12.98075227159389, 13.39182472169175