| L(s) = 1 | − 3.53·3-s − 31.5·7-s − 14.4·9-s + 32.9·11-s + 18.4·13-s + 26.8·17-s − 142.·19-s + 111.·21-s − 147.·23-s + 146.·27-s + 62.9·29-s + 46.1·31-s − 116.·33-s − 276.·37-s − 65.3·39-s − 198.·41-s − 482.·43-s − 190.·47-s + 653.·49-s − 94.9·51-s − 655.·53-s + 505.·57-s − 690.·59-s + 358.·61-s + 457.·63-s − 835.·67-s + 521.·69-s + ⋯ |
| L(s) = 1 | − 0.680·3-s − 1.70·7-s − 0.536·9-s + 0.904·11-s + 0.393·13-s + 0.382·17-s − 1.72·19-s + 1.16·21-s − 1.33·23-s + 1.04·27-s + 0.403·29-s + 0.267·31-s − 0.615·33-s − 1.22·37-s − 0.268·39-s − 0.756·41-s − 1.71·43-s − 0.592·47-s + 1.90·49-s − 0.260·51-s − 1.70·53-s + 1.17·57-s − 1.52·59-s + 0.751·61-s + 0.914·63-s − 1.52·67-s + 0.909·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3494834204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3494834204\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.53T + 27T^{2} \) |
| 7 | \( 1 + 31.5T + 343T^{2} \) |
| 11 | \( 1 - 32.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 62.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 482.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 655.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 690.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 835.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 727.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 524.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 830.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 910.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.801341550095727871757700416663, −8.194439322388693842437021055895, −6.80345242049998514622698620966, −6.36244630894714305910220293210, −5.96865913661828279937243503771, −4.78332380033491613257606395750, −3.73674811103045950170020024244, −3.09017551964739601058171604934, −1.75810215067893134695604704614, −0.27159863422842868020285204338,
0.27159863422842868020285204338, 1.75810215067893134695604704614, 3.09017551964739601058171604934, 3.73674811103045950170020024244, 4.78332380033491613257606395750, 5.96865913661828279937243503771, 6.36244630894714305910220293210, 6.80345242049998514622698620966, 8.194439322388693842437021055895, 8.801341550095727871757700416663
