| L(s) = 1 | − 3-s + 4·7-s − 2·9-s + 13-s − 5·17-s − 6·19-s − 4·21-s − 23-s − 5·25-s + 5·27-s + 29-s + 6·37-s − 39-s + 4·41-s + 43-s + 6·47-s + 9·49-s + 5·51-s − 53-s + 6·57-s + 4·59-s − 61-s − 8·63-s + 69-s + 10·71-s − 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.277·13-s − 1.21·17-s − 1.37·19-s − 0.872·21-s − 0.208·23-s − 25-s + 0.962·27-s + 0.185·29-s + 0.986·37-s − 0.160·39-s + 0.624·41-s + 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.700·51-s − 0.137·53-s + 0.794·57-s + 0.520·59-s − 0.128·61-s − 1.00·63-s + 0.120·69-s + 1.18·71-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.475754623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475754623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.110896978550320552662253128472, −7.42819486533161227403721792372, −6.39615319627986706205775688981, −6.00048241163195681223930704393, −5.12528861943976162291841392643, −4.51577382770578771422186809486, −3.88961928418048064130977000065, −2.48421765811942561901615172141, −1.91813330460733743101011297206, −0.64150514988932763671191576277,
0.64150514988932763671191576277, 1.91813330460733743101011297206, 2.48421765811942561901615172141, 3.88961928418048064130977000065, 4.51577382770578771422186809486, 5.12528861943976162291841392643, 6.00048241163195681223930704393, 6.39615319627986706205775688981, 7.42819486533161227403721792372, 8.110896978550320552662253128472
