| L(s) = 1 | − 2·2-s − 3.39·3-s + 4·4-s + 2.94·5-s + 6.79·6-s − 8·8-s − 15.4·9-s − 5.88·10-s + 7.18·11-s − 13.5·12-s − 35.5·13-s − 9.99·15-s + 16·16-s − 89.7·17-s + 30.9·18-s − 40.9·19-s + 11.7·20-s − 14.3·22-s + 23·23-s + 27.1·24-s − 116.·25-s + 71.0·26-s + 144.·27-s − 231.·29-s + 19.9·30-s + 110.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.653·3-s + 0.5·4-s + 0.263·5-s + 0.462·6-s − 0.353·8-s − 0.573·9-s − 0.186·10-s + 0.197·11-s − 0.326·12-s − 0.757·13-s − 0.171·15-s + 0.250·16-s − 1.28·17-s + 0.405·18-s − 0.494·19-s + 0.131·20-s − 0.139·22-s + 0.208·23-s + 0.231·24-s − 0.930·25-s + 0.535·26-s + 1.02·27-s − 1.48·29-s + 0.121·30-s + 0.638·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3260133939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3260133939\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 3.39T + 27T^{2} \) |
| 5 | \( 1 - 2.94T + 125T^{2} \) |
| 11 | \( 1 - 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 231.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 206.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 503.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 113.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 759.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 565.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 311.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 817.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 325.T + 9.12e5T^{2} \) |
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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
−8.722919514304358952421349460388, −8.038339225549199606206652871231, −6.98368633822266722984578692967, −6.50515231678961706370287415639, −5.60907012166551082336841100133, −4.90253945786082916041625616364, −3.74755013513557145157726711667, −2.53175944401139099158262797595, −1.73606600656466193643962849693, −0.28301859113048722292036697510,
0.28301859113048722292036697510, 1.73606600656466193643962849693, 2.53175944401139099158262797595, 3.74755013513557145157726711667, 4.90253945786082916041625616364, 5.60907012166551082336841100133, 6.50515231678961706370287415639, 6.98368633822266722984578692967, 8.038339225549199606206652871231, 8.722919514304358952421349460388
