| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 6·11-s + 13-s + 3·21-s − 23-s − 5·25-s + 9·27-s − 3·29-s + 3·31-s + 18·33-s − 8·37-s + 3·39-s + 9·41-s − 4·43-s − 13·47-s + 49-s + 4·53-s − 4·59-s + 2·61-s + 6·63-s + 4·67-s − 3·69-s + 5·71-s + 3·73-s − 15·75-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 1.80·11-s + 0.277·13-s + 0.654·21-s − 0.208·23-s − 25-s + 1.73·27-s − 0.557·29-s + 0.538·31-s + 3.13·33-s − 1.31·37-s + 0.480·39-s + 1.40·41-s − 0.609·43-s − 1.89·47-s + 1/7·49-s + 0.549·53-s − 0.520·59-s + 0.256·61-s + 0.755·63-s + 0.488·67-s − 0.361·69-s + 0.593·71-s + 0.351·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.050407702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.050407702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.691305874019703324611083771407, −8.397697726273319604019150386971, −7.48914087784297151131164080191, −6.83363163227438152663054907313, −5.92628684258080595444148818043, −4.57871756056957008057739349846, −3.86193491691139779254847461599, −3.29244781199245755043884169831, −2.08380664141095374617809237015, −1.39241811796677333435466144135,
1.39241811796677333435466144135, 2.08380664141095374617809237015, 3.29244781199245755043884169831, 3.86193491691139779254847461599, 4.57871756056957008057739349846, 5.92628684258080595444148818043, 6.83363163227438152663054907313, 7.48914087784297151131164080191, 8.397697726273319604019150386971, 8.691305874019703324611083771407
