| L(s) = 1 | − 4.61·2-s + 3.68·3-s + 13.2·4-s + 19.7·5-s − 17.0·6-s − 18.7·7-s − 24.4·8-s − 13.4·9-s − 91.3·10-s − 11·11-s + 49.0·12-s + 86.6·14-s + 72.9·15-s + 6.37·16-s + 67.7·17-s + 61.8·18-s + 129.·19-s + 263.·20-s − 69.2·21-s + 50.7·22-s − 103.·23-s − 90.0·24-s + 266.·25-s − 148.·27-s − 249.·28-s + 269.·29-s − 336.·30-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.709·3-s + 1.66·4-s + 1.77·5-s − 1.15·6-s − 1.01·7-s − 1.07·8-s − 0.496·9-s − 2.88·10-s − 0.301·11-s + 1.17·12-s + 1.65·14-s + 1.25·15-s + 0.0996·16-s + 0.966·17-s + 0.809·18-s + 1.56·19-s + 2.94·20-s − 0.719·21-s + 0.491·22-s − 0.939·23-s − 0.766·24-s + 2.13·25-s − 1.06·27-s − 1.68·28-s + 1.72·29-s − 2.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.572705871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572705871\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.61T + 8T^{2} \) |
| 3 | \( 1 - 3.68T + 27T^{2} \) |
| 5 | \( 1 - 19.7T + 125T^{2} \) |
| 7 | \( 1 + 18.7T + 343T^{2} \) |
| 17 | \( 1 - 67.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.87T + 5.06e4T^{2} \) |
| 41 | \( 1 - 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 717.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 583.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 697.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 620.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 790.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 98.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 455.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 346.T + 9.12e5T^{2} \) |
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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
−9.108939963089089363957698876043, −8.278506532389690864890214910996, −7.63606629161558081227142634857, −6.56740048114893946252457780983, −6.05390990170726460860480019002, −5.12768840988314357428694763279, −3.19222334354108655215839227455, −2.67410363336006479736093257359, −1.71723179540143207177062645495, −0.73264599778494164027182575035,
0.73264599778494164027182575035, 1.71723179540143207177062645495, 2.67410363336006479736093257359, 3.19222334354108655215839227455, 5.12768840988314357428694763279, 6.05390990170726460860480019002, 6.56740048114893946252457780983, 7.63606629161558081227142634857, 8.278506532389690864890214910996, 9.108939963089089363957698876043
