| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·11-s + 2·13-s + 15-s − 8·17-s + 8·19-s + 2·21-s + 8·23-s + 25-s − 27-s − 10·29-s − 8·31-s + 2·33-s + 2·35-s − 2·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s − 4·47-s − 3·49-s + 8·51-s + 2·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 1.94·17-s + 1.83·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.348·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 1.12·51-s + 0.274·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8628727221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8628727221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.638551538442009893577523164070, −7.42453918815745026023592410788, −7.14082162084889437091737778969, −6.31312099042302818818590735279, −5.41051129030844572729002610372, −4.89112307430868275798271944931, −3.75282749942956815280684885618, −3.17592623893454142168104729943, −1.92860669592669570819603903176, −0.53824953394909956001202651788,
0.53824953394909956001202651788, 1.92860669592669570819603903176, 3.17592623893454142168104729943, 3.75282749942956815280684885618, 4.89112307430868275798271944931, 5.41051129030844572729002610372, 6.31312099042302818818590735279, 7.14082162084889437091737778969, 7.42453918815745026023592410788, 8.638551538442009893577523164070
