| L(s) = 1 | − i·2-s + 1.30i·3-s − 4-s + 1.30·6-s + 4.30i·7-s + i·8-s + 1.30·9-s − 4.60·11-s − 1.30i·12-s − 2.69i·13-s + 4.30·14-s + 16-s + 6.90i·17-s − 1.30i·18-s − 6.60·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.752i·3-s − 0.5·4-s + 0.531·6-s + 1.62i·7-s + 0.353i·8-s + 0.434·9-s − 1.38·11-s − 0.376i·12-s − 0.748i·13-s + 1.14·14-s + 0.250·16-s + 1.67i·17-s − 0.307i·18-s − 1.51·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5774360807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5774360807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30iT - 3T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 + 2.69iT - 13T^{2} \) |
| 17 | \( 1 - 6.90iT - 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 5.30iT - 23T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 - 0.302iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6.90iT - 53T^{2} \) |
| 59 | \( 1 + 9.90T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.21iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 5.90T + 79T^{2} \) |
| 83 | \( 1 - 1.39iT - 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 11.9iT - 97T^{2} \) |
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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
−10.14000195351516464434825688434, −9.052520177161929716463900246121, −8.521325323564315159530148354193, −7.83771350317401000524105998336, −6.27648239971479578216313385115, −5.56463385151346659304099688097, −4.76402768615726353439358678894, −3.86455569611502057192139370924, −2.71698539084578991462752636682, −2.00474466670968750336781631909,
0.22309634533403511021705049936, 1.56828002747102530951880173100, 3.04975860550152159782759450250, 4.40830671801111321869251410726, 4.82344170162601018143572561715, 6.23037465388274450395052809983, 6.88479421719605027711145843024, 7.59125453342037059970491332928, 7.86068959825153542787629979509, 9.091567762600670914885438000415
