| L(s) = 1 | − 3.46·2-s + 3.97·4-s + 5·5-s + 17.6·7-s + 13.9·8-s − 17.3·10-s − 37.9·11-s + 69.0·13-s − 61.1·14-s − 79.9·16-s + 126.·17-s + 11.1·19-s + 19.8·20-s + 131.·22-s + 23·23-s + 25·25-s − 238.·26-s + 70.2·28-s − 109.·29-s + 210.·31-s + 165.·32-s − 437.·34-s + 88.3·35-s + 436.·37-s − 38.7·38-s + 69.6·40-s + 132.·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.496·4-s + 0.447·5-s + 0.954·7-s + 0.615·8-s − 0.547·10-s − 1.03·11-s + 1.47·13-s − 1.16·14-s − 1.24·16-s + 1.80·17-s + 0.135·19-s + 0.222·20-s + 1.27·22-s + 0.208·23-s + 0.200·25-s − 1.80·26-s + 0.474·28-s − 0.702·29-s + 1.21·31-s + 0.913·32-s − 2.20·34-s + 0.426·35-s + 1.94·37-s − 0.165·38-s + 0.275·40-s + 0.505·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.481717913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481717913\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 3.46T + 8T^{2} \) |
| 7 | \( 1 - 17.6T + 343T^{2} \) |
| 11 | \( 1 + 37.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 347.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
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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
−9.582281336055592187881628986573, −8.653054204123174077433989533128, −7.938635338928957048481784903231, −7.59885311235206324120066146947, −6.18201189549397271531044842880, −5.35651003597376439022555275407, −4.34688947879824989638240852916, −2.92379543322230420286904726480, −1.58352225995218499647825430731, −0.865018489391644947576306318407,
0.865018489391644947576306318407, 1.58352225995218499647825430731, 2.92379543322230420286904726480, 4.34688947879824989638240852916, 5.35651003597376439022555275407, 6.18201189549397271531044842880, 7.59885311235206324120066146947, 7.938635338928957048481784903231, 8.653054204123174077433989533128, 9.582281336055592187881628986573
