| L(s) = 1 | + 3-s + 3.46·5-s − 3.46·7-s + 9-s + 3.46·15-s + 6·17-s + 4·19-s − 3.46·21-s + 6.92·23-s + 6.99·25-s + 27-s − 3.46·29-s − 3.46·31-s − 11.9·35-s − 6.92·37-s + 6·41-s + 4·43-s + 3.46·45-s − 6.92·47-s + 4.99·49-s + 6·51-s − 3.46·53-s + 4·57-s − 12·59-s + 6.92·61-s − 3.46·63-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54·5-s − 1.30·7-s + 0.333·9-s + 0.894·15-s + 1.45·17-s + 0.917·19-s − 0.755·21-s + 1.44·23-s + 1.39·25-s + 0.192·27-s − 0.643·29-s − 0.622·31-s − 2.02·35-s − 1.13·37-s + 0.937·41-s + 0.609·43-s + 0.516·45-s − 1.01·47-s + 0.714·49-s + 0.840·51-s − 0.475·53-s + 0.529·57-s − 1.56·59-s + 0.887·61-s − 0.436·63-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.242163661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.242163661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.01708505222310147168735736375, −9.465714208982566595750229259561, −9.039980779626859621063654855052, −7.64586829684534494324413764162, −6.79706933710280097510253097845, −5.88535346965831285894124540745, −5.17519834502749886916405280921, −3.46641820802356441166979549669, −2.79523610059826734126120834496, −1.40376133463709272202370477837,
1.40376133463709272202370477837, 2.79523610059826734126120834496, 3.46641820802356441166979549669, 5.17519834502749886916405280921, 5.88535346965831285894124540745, 6.79706933710280097510253097845, 7.64586829684534494324413764162, 9.039980779626859621063654855052, 9.465714208982566595750229259561, 10.01708505222310147168735736375
