| L(s) = 1 | − 8.19·3-s + 21.7·7-s + 40.0·9-s − 37.8·11-s − 52.7·13-s + 99.9·17-s − 116.·19-s − 177.·21-s + 37.7·23-s − 107.·27-s + 218.·29-s + 67.3·31-s + 310.·33-s + 362.·37-s + 431.·39-s − 291.·41-s + 183.·43-s − 443.·47-s + 128.·49-s − 818.·51-s + 416.·53-s + 956.·57-s − 828.·59-s + 442.·61-s + 870.·63-s + 570.·67-s − 309.·69-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 1.17·7-s + 1.48·9-s − 1.03·11-s − 1.12·13-s + 1.42·17-s − 1.40·19-s − 1.84·21-s + 0.342·23-s − 0.764·27-s + 1.40·29-s + 0.390·31-s + 1.63·33-s + 1.61·37-s + 1.77·39-s − 1.11·41-s + 0.649·43-s − 1.37·47-s + 0.373·49-s − 2.24·51-s + 1.07·53-s + 2.22·57-s − 1.82·59-s + 0.929·61-s + 1.74·63-s + 1.03·67-s − 0.539·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8.19T + 27T^{2} \) |
| 7 | \( 1 - 21.7T + 343T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 362.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 443.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 828.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 442.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 570.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 341.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 426.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 982.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 926.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.88e3T + 9.12e5T^{2} \) |
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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
−9.896816890344427104216719445391, −8.346025995525180234052312379169, −7.70873546053020332219537387422, −6.71384403428448125044097406727, −5.73513794625829758251571700241, −4.98855514740916866330555528220, −4.48876818106836386834661491442, −2.62220465243881423776022596254, −1.21490729488452548882484097898, 0,
1.21490729488452548882484097898, 2.62220465243881423776022596254, 4.48876818106836386834661491442, 4.98855514740916866330555528220, 5.73513794625829758251571700241, 6.71384403428448125044097406727, 7.70873546053020332219537387422, 8.346025995525180234052312379169, 9.896816890344427104216719445391
