| L(s) = 1 | + 2i·2-s + 2i·3-s − 4·4-s + (5 + 10i)5-s − 4·6-s − 8i·8-s + 23·9-s + (−20 + 10i)10-s − 28·11-s − 8i·12-s + 12i·13-s + (−20 + 10i)15-s + 16·16-s + 64i·17-s + 46i·18-s − 60·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.384i·3-s − 0.5·4-s + (0.447 + 0.894i)5-s − 0.272·6-s − 0.353i·8-s + 0.851·9-s + (−0.632 + 0.316i)10-s − 0.767·11-s − 0.192i·12-s + 0.256i·13-s + (−0.344 + 0.172i)15-s + 0.250·16-s + 0.913i·17-s + 0.602i·18-s − 0.724·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.273234713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273234713\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (-5 - 10i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 11 | \( 1 + 28T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 60T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 242T + 6.89e4T^{2} \) |
| 43 | \( 1 + 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 20T + 2.05e5T^{2} \) |
| 61 | \( 1 + 542T + 2.26e5T^{2} \) |
| 67 | \( 1 + 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 720T + 4.93e5T^{2} \) |
| 83 | \( 1 + 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 490T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3iT - 9.12e5T^{2} \) |
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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
−10.59677049081958026467992305107, −10.28310959623063488724665626772, −9.336102705172220950322566356252, −8.250374680932885662051298296313, −7.29009788234279204657108503817, −6.51933442468180005342740096683, −5.57309162295232177041269287641, −4.45588919613803919402442398671, −3.36703067994750769363022655717, −1.83979462530784620400123479414,
0.39010393057023164507448278382, 1.59362054481535588627025835858, 2.69679450382151762891425293098, 4.28763849262282265775473743920, 5.04891229892299965853446587551, 6.19042967336081089797820315698, 7.47186657169440413599139992438, 8.338587278324295044867447015589, 9.309622555126909319751524018419, 10.04436436709044549703219310533
