| L(s) = 1 | + 0.197·2-s − 0.963·3-s − 7.96·4-s − 5·5-s − 0.190·6-s + 0.480·7-s − 3.14·8-s − 26.0·9-s − 0.986·10-s + 11·11-s + 7.67·12-s − 92.4·13-s + 0.0947·14-s + 4.81·15-s + 63.0·16-s − 100.·17-s − 5.14·18-s + 19·19-s + 39.8·20-s − 0.462·21-s + 2.17·22-s − 183.·23-s + 3.03·24-s + 25·25-s − 18.2·26-s + 51.1·27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | + 0.0697·2-s − 0.185·3-s − 0.995·4-s − 0.447·5-s − 0.0129·6-s + 0.0259·7-s − 0.139·8-s − 0.965·9-s − 0.0312·10-s + 0.301·11-s + 0.184·12-s − 1.97·13-s + 0.00180·14-s + 0.0829·15-s + 0.985·16-s − 1.42·17-s − 0.0673·18-s + 0.229·19-s + 0.445·20-s − 0.00480·21-s + 0.0210·22-s − 1.66·23-s + 0.0258·24-s + 0.200·25-s − 0.137·26-s + 0.364·27-s − 0.0257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1683915387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1683915387\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 0.197T + 8T^{2} \) |
| 3 | \( 1 + 0.963T + 27T^{2} \) |
| 7 | \( 1 - 0.480T + 343T^{2} \) |
| 13 | \( 1 + 92.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 100.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 499.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 565.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 485.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 328.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 11.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 52.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 231.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 477.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 226.T + 9.12e5T^{2} \) |
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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
−9.591588313620249424061860995945, −8.648030683687716680430296889380, −8.060581331581944813548485003047, −7.07023523927955769720626908185, −6.02326983188027163728408721349, −5.01570708594347917298621219675, −4.44641291749851652986600754498, −3.34254510385287372352633505091, −2.13801574329732610143426048779, −0.20454475522419233132252162827,
0.20454475522419233132252162827, 2.13801574329732610143426048779, 3.34254510385287372352633505091, 4.44641291749851652986600754498, 5.01570708594347917298621219675, 6.02326983188027163728408721349, 7.07023523927955769720626908185, 8.060581331581944813548485003047, 8.648030683687716680430296889380, 9.591588313620249424061860995945
