L(s) = 1 | + 0.732·2-s − 7.46·4-s − 5·5-s − 6.92·7-s − 11.3·8-s − 3.66·10-s − 37.4·11-s − 38.9·13-s − 5.07·14-s + 51.4·16-s + 80.9·17-s + 112.·19-s + 37.3·20-s − 27.4·22-s − 13.4·23-s + 25·25-s − 28.4·26-s + 51.7·28-s + 43.4·29-s − 149.·31-s + 128.·32-s + 59.2·34-s + 34.6·35-s − 218.·37-s + 82.5·38-s + 56.6·40-s + 372.·41-s + ⋯ |
L(s) = 1 | + 0.258·2-s − 0.933·4-s − 0.447·5-s − 0.374·7-s − 0.500·8-s − 0.115·10-s − 1.02·11-s − 0.830·13-s − 0.0968·14-s + 0.803·16-s + 1.15·17-s + 1.36·19-s + 0.417·20-s − 0.266·22-s − 0.121·23-s + 0.200·25-s − 0.214·26-s + 0.349·28-s + 0.278·29-s − 0.867·31-s + 0.708·32-s + 0.299·34-s + 0.167·35-s − 0.970·37-s + 0.352·38-s + 0.223·40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.120472273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120472273\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 2 | \( 1 - 0.732T + 8T^{2} \) |
| 7 | \( 1 + 6.92T + 343T^{2} \) |
| 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 460.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 214.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 445.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 401.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 1.44T + 2.26e5T^{2} \) |
| 67 | \( 1 + 816.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 147.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 432.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 513T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−10.71219876121537935078554994302, −9.841003726026216146639662146285, −9.139696903754866133881869430916, −7.925054545197753426535918715729, −7.35230830142601247188023136304, −5.69146875848402258797863175685, −5.06898023368518580619222742805, −3.83652209701108997212525189487, −2.83311436785666374851377459104, −0.65566503017852838083783047216,
0.65566503017852838083783047216, 2.83311436785666374851377459104, 3.83652209701108997212525189487, 5.06898023368518580619222742805, 5.69146875848402258797863175685, 7.35230830142601247188023136304, 7.925054545197753426535918715729, 9.139696903754866133881869430916, 9.841003726026216146639662146285, 10.71219876121537935078554994302