| L(s) = 1 | + 1.66·3-s − 5-s − 3.21·7-s − 0.214·9-s − 5.33·11-s + 4.57·13-s − 1.66·15-s + 4.12·17-s + 6.24·19-s − 5.36·21-s + 8.57·23-s + 25-s − 5.36·27-s − 29-s − 0.123·31-s − 8.90·33-s + 3.21·35-s + 4.42·37-s + 7.64·39-s − 8.42·41-s − 6.12·43-s + 0.214·45-s + 10.6·47-s + 3.33·49-s + 6.88·51-s + 12.3·53-s + 5.33·55-s + ⋯ |
| L(s) = 1 | + 0.963·3-s − 0.447·5-s − 1.21·7-s − 0.0713·9-s − 1.60·11-s + 1.26·13-s − 0.430·15-s + 1.00·17-s + 1.43·19-s − 1.17·21-s + 1.78·23-s + 0.200·25-s − 1.03·27-s − 0.185·29-s − 0.0222·31-s − 1.55·33-s + 0.543·35-s + 0.728·37-s + 1.22·39-s − 1.31·41-s − 0.933·43-s + 0.0319·45-s + 1.55·47-s + 0.475·49-s + 0.963·51-s + 1.69·53-s + 0.719·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.883227332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883227332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 8.57T + 23T^{2} \) |
| 31 | \( 1 + 0.123T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 - 5.58T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 0.123T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 + 0.578T + 97T^{2} \) |
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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
−8.866703465591188351439423311054, −8.309960388143332514419635534547, −7.54530272852871370740692503834, −6.92233112995759935924931077904, −5.72750182299157821476192970761, −5.19320440104876536113294931773, −3.64399513893312667199312088593, −3.25779977367364136479971634168, −2.57250761254618309467004292437, −0.838717445333176573837691876508,
0.838717445333176573837691876508, 2.57250761254618309467004292437, 3.25779977367364136479971634168, 3.64399513893312667199312088593, 5.19320440104876536113294931773, 5.72750182299157821476192970761, 6.92233112995759935924931077904, 7.54530272852871370740692503834, 8.309960388143332514419635534547, 8.866703465591188351439423311054
