| L(s) = 1 | + 3.27·2-s + 2.75·4-s − 17.5·5-s − 26.6·7-s − 17.2·8-s − 57.5·10-s + 21.4·11-s − 87.5·14-s − 78.4·16-s − 83.9·17-s − 77.1·19-s − 48.2·20-s + 70.1·22-s − 142.·23-s + 182.·25-s − 73.4·28-s − 134.·29-s + 122.·31-s − 119.·32-s − 275.·34-s + 468.·35-s − 222.·37-s − 252.·38-s + 301.·40-s − 198.·41-s + 154.·43-s + 58.8·44-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.343·4-s − 1.56·5-s − 1.44·7-s − 0.760·8-s − 1.81·10-s + 0.586·11-s − 1.67·14-s − 1.22·16-s − 1.19·17-s − 0.931·19-s − 0.539·20-s + 0.680·22-s − 1.28·23-s + 1.46·25-s − 0.495·28-s − 0.859·29-s + 0.710·31-s − 0.660·32-s − 1.38·34-s + 2.26·35-s − 0.989·37-s − 1.07·38-s + 1.19·40-s − 0.755·41-s + 0.548·43-s + 0.201·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4561683032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4561683032\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.27T + 8T^{2} \) |
| 5 | \( 1 + 17.5T + 125T^{2} \) |
| 7 | \( 1 + 26.6T + 343T^{2} \) |
| 11 | \( 1 - 21.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 78.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 477.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 496.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 861.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 967.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 591.T + 9.12e5T^{2} \) |
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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
−8.937319616225252577738619913274, −8.400756697122494009309986269529, −7.15908328332836377778253168725, −6.58427770500920710391221159641, −5.84836317641206575181737476401, −4.58367664639083706638737871229, −3.92441127799889548323670646328, −3.53456508788099116756960026123, −2.41592038544018995897247183873, −0.25675958354240277807929701429,
0.25675958354240277807929701429, 2.41592038544018995897247183873, 3.53456508788099116756960026123, 3.92441127799889548323670646328, 4.58367664639083706638737871229, 5.84836317641206575181737476401, 6.58427770500920710391221159641, 7.15908328332836377778253168725, 8.400756697122494009309986269529, 8.937319616225252577738619913274
