L(s) = 1 | + 5-s + 7-s − 6·11-s − 13-s + 19-s − 6·23-s + 25-s + 6·29-s − 8·31-s + 35-s − 7·37-s − 6·41-s + 4·43-s − 12·47-s − 6·49-s − 6·53-s − 6·55-s + 11·61-s − 65-s + 7·67-s + 6·71-s + 11·73-s − 6·77-s + 79-s − 6·83-s − 12·89-s − 91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.15·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 6/7·49-s − 0.824·53-s − 0.809·55-s + 1.40·61-s − 0.124·65-s + 0.855·67-s + 0.712·71-s + 1.28·73-s − 0.683·77-s + 0.112·79-s − 0.658·83-s − 1.27·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.453757101212738665043125480157, −8.054815367760551473988773648865, −7.20149128213989535337754984544, −6.30275477453639231316674547503, −5.26056570635904728739855325749, −4.99617429415613386423394911674, −3.65778919398785502430721530689, −2.61978163780340119595385627704, −1.76173777505214450808543092451, 0,
1.76173777505214450808543092451, 2.61978163780340119595385627704, 3.65778919398785502430721530689, 4.99617429415613386423394911674, 5.26056570635904728739855325749, 6.30275477453639231316674547503, 7.20149128213989535337754984544, 8.054815367760551473988773648865, 8.453757101212738665043125480157