| L(s) = 1 | + 3-s − 3·5-s + 5·7-s + 9-s − 11-s + 2·13-s − 3·15-s − 17-s + 19-s + 5·21-s + 4·23-s + 4·25-s + 27-s − 2·29-s + 6·31-s − 33-s − 15·35-s + 2·39-s + 43-s − 3·45-s + 9·47-s + 18·49-s − 51-s + 10·53-s + 3·55-s + 57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.774·15-s − 0.242·17-s + 0.229·19-s + 1.09·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.174·33-s − 2.53·35-s + 0.320·39-s + 0.152·43-s − 0.447·45-s + 1.31·47-s + 18/7·49-s − 0.140·51-s + 1.37·53-s + 0.404·55-s + 0.132·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918329503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918329503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.24757512594894707132035038390, −8.794614827642349780603070220632, −8.439333560047520544649775927937, −7.66505986486694333159066241147, −7.12629091377271134552327540430, −5.52282167690075554303727517698, −4.53894929342930739368620916750, −3.92764554574875840306496068176, −2.61182434413499280414159715388, −1.19093389013722883778513109873,
1.19093389013722883778513109873, 2.61182434413499280414159715388, 3.92764554574875840306496068176, 4.53894929342930739368620916750, 5.52282167690075554303727517698, 7.12629091377271134552327540430, 7.66505986486694333159066241147, 8.439333560047520544649775927937, 8.794614827642349780603070220632, 10.24757512594894707132035038390
