| L(s) = 1 | + 2·11-s − 2·13-s + 4·17-s + 19-s + 2·23-s − 5·25-s + 6·29-s + 4·31-s + 2·37-s − 2·41-s + 4·43-s + 10·47-s − 7·49-s + 6·53-s + 4·59-s + 2·61-s + 4·67-s + 4·71-s − 6·73-s + 8·79-s − 2·83-s + 6·89-s − 10·97-s + 4·101-s + 12·107-s − 2·109-s + 6·113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 0.554·13-s + 0.970·17-s + 0.229·19-s + 0.417·23-s − 25-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.45·47-s − 49-s + 0.824·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.474·71-s − 0.702·73-s + 0.900·79-s − 0.219·83-s + 0.635·89-s − 1.01·97-s + 0.398·101-s + 1.16·107-s − 0.191·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742682731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742682731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.702234765741361693683284102680, −8.799684961473349886462534346259, −7.949331149426297835742684058956, −7.19538412605426330134786410439, −6.30049703135965645039602556044, −5.43558490585074567595679139136, −4.48299372830498550967126311807, −3.50474201494392624269705999205, −2.41633770323409374987205582753, −1.00307307577357426963677328458,
1.00307307577357426963677328458, 2.41633770323409374987205582753, 3.50474201494392624269705999205, 4.48299372830498550967126311807, 5.43558490585074567595679139136, 6.30049703135965645039602556044, 7.19538412605426330134786410439, 7.949331149426297835742684058956, 8.799684961473349886462534346259, 9.702234765741361693683284102680
