| L(s) = 1 | + 3·5-s + 7-s + 3·11-s + 2·13-s − 6·17-s + 5·19-s − 9·23-s + 4·25-s + 6·29-s − 31-s + 3·35-s + 11·37-s + 3·41-s − 4·43-s − 12·47-s + 49-s + 9·55-s + 8·61-s + 6·65-s − 10·67-s + 3·71-s + 8·73-s + 3·77-s − 4·79-s + 6·83-s − 18·85-s + 3·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 1.45·17-s + 1.14·19-s − 1.87·23-s + 4/5·25-s + 1.11·29-s − 0.179·31-s + 0.507·35-s + 1.80·37-s + 0.468·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.21·55-s + 1.02·61-s + 0.744·65-s − 1.22·67-s + 0.356·71-s + 0.936·73-s + 0.341·77-s − 0.450·79-s + 0.658·83-s − 1.95·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.081849511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.081849511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.13907841601863165967919400748, −9.570908741228429428937688272533, −8.780337299127953556708917529495, −7.84864688607256323547159673807, −6.49821726406191861904352357423, −6.13420642981607383189262446190, −4.99058996855721417798172650794, −3.94315135306159377097127451803, −2.44662848712626536551871227031, −1.41871132954067485584883752975,
1.41871132954067485584883752975, 2.44662848712626536551871227031, 3.94315135306159377097127451803, 4.99058996855721417798172650794, 6.13420642981607383189262446190, 6.49821726406191861904352357423, 7.84864688607256323547159673807, 8.780337299127953556708917529495, 9.570908741228429428937688272533, 10.13907841601863165967919400748
