| L(s) = 1 | − 2·4-s + 7·13-s + 4·16-s + 7·19-s − 5·25-s + 7·31-s − 37-s + 5·43-s − 14·52-s − 14·61-s − 8·64-s + 11·67-s + 7·73-s − 14·76-s − 13·79-s − 14·97-s + 10·100-s + 7·103-s + 17·109-s + ⋯ |
| L(s) = 1 | − 4-s + 1.94·13-s + 16-s + 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s + 0.762·43-s − 1.94·52-s − 1.79·61-s − 64-s + 1.34·67-s + 0.819·73-s − 1.60·76-s − 1.46·79-s − 1.42·97-s + 100-s + 0.689·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209177364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209177364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.12611090306175738214534717990, −10.09209954751813487013591320496, −9.291988044548486732019479222726, −8.461373500657640359204806994178, −7.67458414128501773545216406647, −6.24141795466275696941914757951, −5.40791154265456502263047742620, −4.19234808180063134723072414533, −3.28050491254919886424015350098, −1.14668647924786773157165026874,
1.14668647924786773157165026874, 3.28050491254919886424015350098, 4.19234808180063134723072414533, 5.40791154265456502263047742620, 6.24141795466275696941914757951, 7.67458414128501773545216406647, 8.461373500657640359204806994178, 9.291988044548486732019479222726, 10.09209954751813487013591320496, 11.12611090306175738214534717990
