| L(s) = 1 | + 3.31·3-s − 5-s − 4.86·7-s + 8.02·9-s + 4.15·11-s + 2.66·13-s − 3.31·15-s − 2.86·17-s + 0.651·19-s − 16.1·21-s + 4.65·23-s + 25-s + 16.6·27-s + 29-s + 4.17·31-s + 13.7·33-s + 4.86·35-s + 11.3·37-s + 8.85·39-s + 3.83·41-s − 12.0·43-s − 8.02·45-s − 7.48·47-s + 16.7·49-s − 9.52·51-s + 1.61·53-s − 4.15·55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 0.447·5-s − 1.84·7-s + 2.67·9-s + 1.25·11-s + 0.739·13-s − 0.857·15-s − 0.695·17-s + 0.149·19-s − 3.52·21-s + 0.970·23-s + 0.200·25-s + 3.20·27-s + 0.185·29-s + 0.749·31-s + 2.39·33-s + 0.823·35-s + 1.86·37-s + 1.41·39-s + 0.599·41-s − 1.83·43-s − 1.19·45-s − 1.09·47-s + 2.38·49-s − 1.33·51-s + 0.222·53-s − 0.559·55-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
| \(L(1)\) |
\(\approx\) |
\(3.183452417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.183452417\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.31T + 3T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 - 0.651T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 1.61T + 53T^{2} \) |
| 59 | \( 1 - 3.33T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 + 0.903T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.018738680394113288765886556380, −8.432839481848450285234139885996, −7.56214892926331181619837161746, −6.60863274099841056641377590771, −6.45428396030898889464505239520, −4.56391399622478394105090429628, −3.74801876499061573649718514447, −3.29711895461798390196372033096, −2.51946850000517124872616179491, −1.12622528782494871448205874528,
1.12622528782494871448205874528, 2.51946850000517124872616179491, 3.29711895461798390196372033096, 3.74801876499061573649718514447, 4.56391399622478394105090429628, 6.45428396030898889464505239520, 6.60863274099841056641377590771, 7.56214892926331181619837161746, 8.432839481848450285234139885996, 9.018738680394113288765886556380
