| L(s) = 1 | + 5.36·3-s + 12.6·5-s + 7·7-s + 1.83·9-s + 10.9·11-s + 28.7·13-s + 67.8·15-s − 17·17-s − 52.6·19-s + 37.5·21-s + 160.·23-s + 34.4·25-s − 135.·27-s + 177.·29-s − 79.7·31-s + 59.0·33-s + 88.3·35-s + 214.·37-s + 154.·39-s + 473.·41-s − 546.·43-s + 23.1·45-s + 82.8·47-s + 49·49-s − 91.2·51-s + 280.·53-s + 138.·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.12·5-s + 0.377·7-s + 0.0679·9-s + 0.301·11-s + 0.613·13-s + 1.16·15-s − 0.242·17-s − 0.635·19-s + 0.390·21-s + 1.45·23-s + 0.275·25-s − 0.963·27-s + 1.13·29-s − 0.462·31-s + 0.311·33-s + 0.426·35-s + 0.953·37-s + 0.634·39-s + 1.80·41-s − 1.93·43-s + 0.0767·45-s + 0.257·47-s + 0.142·49-s − 0.250·51-s + 0.726·53-s + 0.340·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.693829352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.693829352\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 5.36T + 27T^{2} \) |
| 5 | \( 1 - 12.6T + 125T^{2} \) |
| 11 | \( 1 - 10.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 52.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 473.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 546.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 82.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 280.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 735.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 406.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 208.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 169.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 32.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 440.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 561.T + 9.12e5T^{2} \) |
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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
−8.795687195012769705188053276410, −8.373907039433031478102676138737, −7.34804045466139195122759442025, −6.44634070351632262227578395724, −5.72819290207289375425298555755, −4.76215408660541630799674783874, −3.73394854629441453777178030309, −2.72932699718495149346219153182, −2.04724346080965460500249775036, −1.00425627235343515037826009863,
1.00425627235343515037826009863, 2.04724346080965460500249775036, 2.72932699718495149346219153182, 3.73394854629441453777178030309, 4.76215408660541630799674783874, 5.72819290207289375425298555755, 6.44634070351632262227578395724, 7.34804045466139195122759442025, 8.373907039433031478102676138737, 8.795687195012769705188053276410
