| L(s) = 1 | − 9.08·3-s + 6.91·5-s − 7·7-s + 55.4·9-s − 11·11-s + 7.08·13-s − 62.8·15-s − 68.3·17-s + 94.5·19-s + 63.5·21-s − 104.·23-s − 77.1·25-s − 258.·27-s + 48.4·29-s − 38.8·31-s + 99.9·33-s − 48.4·35-s + 144.·37-s − 64.3·39-s + 409.·41-s + 16.8·43-s + 383.·45-s + 535.·47-s + 49·49-s + 620.·51-s − 155.·53-s − 76.0·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s + 0.618·5-s − 0.377·7-s + 2.05·9-s − 0.301·11-s + 0.151·13-s − 1.08·15-s − 0.974·17-s + 1.14·19-s + 0.660·21-s − 0.945·23-s − 0.617·25-s − 1.84·27-s + 0.310·29-s − 0.225·31-s + 0.527·33-s − 0.233·35-s + 0.641·37-s − 0.264·39-s + 1.56·41-s + 0.0597·43-s + 1.27·45-s + 1.66·47-s + 0.142·49-s + 1.70·51-s − 0.403·53-s − 0.186·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 9.08T + 27T^{2} \) |
| 5 | \( 1 - 6.91T + 125T^{2} \) |
| 13 | \( 1 - 7.08T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 38.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 16.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 449.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 529.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 296.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 942.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 87.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.85e3T + 9.12e5T^{2} \) |
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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
−9.350269933833996350062710989618, −7.898209735752830245169748110099, −7.03063144157644431328008941950, −6.13241689884676373680397279397, −5.77734032182741468401181692893, −4.86339502030987505831465778457, −3.95046814275195003704012955176, −2.37078891867964026144243664741, −1.08672168525642017249992662632, 0,
1.08672168525642017249992662632, 2.37078891867964026144243664741, 3.95046814275195003704012955176, 4.86339502030987505831465778457, 5.77734032182741468401181692893, 6.13241689884676373680397279397, 7.03063144157644431328008941950, 7.898209735752830245169748110099, 9.350269933833996350062710989618
