L(s) = 1 | + 2·7-s + 5·11-s − 5·17-s − 5·19-s + 6·23-s − 4·29-s − 10·31-s − 10·37-s − 5·41-s − 4·43-s − 8·47-s − 3·49-s + 10·53-s − 10·61-s − 3·67-s − 5·73-s + 10·77-s − 10·79-s − 83-s + 9·89-s + 10·97-s − 2·101-s + 16·103-s − 3·107-s + 10·109-s − 15·113-s − 10·119-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.50·11-s − 1.21·17-s − 1.14·19-s + 1.25·23-s − 0.742·29-s − 1.79·31-s − 1.64·37-s − 0.780·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 1.28·61-s − 0.366·67-s − 0.585·73-s + 1.13·77-s − 1.12·79-s − 0.109·83-s + 0.953·89-s + 1.01·97-s − 0.199·101-s + 1.57·103-s − 0.290·107-s + 0.957·109-s − 1.41·113-s − 0.916·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.44271046078467485799905527839, −6.84799877638117622237573923452, −6.35814816166339174894520680930, −5.36327154217626596508697279375, −4.70349093523531388575214392256, −3.99054406218724174759783422485, −3.28950523082798615455067725259, −1.96660783652690174666162237215, −1.53879008655992227830958465983, 0,
1.53879008655992227830958465983, 1.96660783652690174666162237215, 3.28950523082798615455067725259, 3.99054406218724174759783422485, 4.70349093523531388575214392256, 5.36327154217626596508697279375, 6.35814816166339174894520680930, 6.84799877638117622237573923452, 7.44271046078467485799905527839