| L(s) = 1 | + 2·5-s − 3·9-s − 2·13-s − 6·17-s + 4·19-s − 8·23-s − 25-s − 2·29-s + 31-s − 10·37-s − 6·41-s + 8·43-s − 6·45-s + 8·47-s − 7·49-s + 6·53-s − 12·59-s + 6·61-s − 4·65-s − 12·67-s − 8·71-s + 10·73-s + 8·79-s + 9·81-s + 8·83-s − 12·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 0.179·31-s − 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.16·47-s − 49-s + 0.824·53-s − 1.56·59-s + 0.768·61-s − 0.496·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 81-s + 0.878·83-s − 1.30·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.911717137174663405595486729622, −8.067419021434777137162075637049, −7.17663415905550465853772545662, −6.23927644667191941192015479418, −5.67007372238293440407484071620, −4.85213555842603099342069383801, −3.73552653057987685997850390510, −2.58717477143022626568369362485, −1.85194090492113304245340785765, 0,
1.85194090492113304245340785765, 2.58717477143022626568369362485, 3.73552653057987685997850390510, 4.85213555842603099342069383801, 5.67007372238293440407484071620, 6.23927644667191941192015479418, 7.17663415905550465853772545662, 8.067419021434777137162075637049, 8.911717137174663405595486729622
