| L(s) = 1 | − 2.37i·2-s + 1.27i·3-s − 3.65·4-s + 3.02·6-s − 0.726i·7-s + 3.92i·8-s + 1.37·9-s − 0.273·11-s − 4.65i·12-s − 5.95i·13-s − 1.72·14-s + 2.02·16-s − 5.27i·17-s − 3.27i·18-s − 19-s + ⋯ |
| L(s) = 1 | − 1.68i·2-s + 0.735i·3-s − 1.82·4-s + 1.23·6-s − 0.274i·7-s + 1.38i·8-s + 0.459·9-s − 0.0825·11-s − 1.34i·12-s − 1.65i·13-s − 0.461·14-s + 0.507·16-s − 1.27i·17-s − 0.771i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.273854 - 1.16006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.273854 - 1.16006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37iT - 2T^{2} \) |
| 3 | \( 1 - 1.27iT - 3T^{2} \) |
| 7 | \( 1 + 0.726iT - 7T^{2} \) |
| 11 | \( 1 + 0.273T + 11T^{2} \) |
| 13 | \( 1 + 5.95iT - 13T^{2} \) |
| 17 | \( 1 + 5.27iT - 17T^{2} \) |
| 23 | \( 1 + 3.67iT - 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 8.12iT - 37T^{2} \) |
| 41 | \( 1 + 9.43T + 41T^{2} \) |
| 43 | \( 1 - 9.81iT - 43T^{2} \) |
| 47 | \( 1 + 12.1iT - 47T^{2} \) |
| 53 | \( 1 - 5.69iT - 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 0.103T + 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 6.67iT - 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 + 0.488iT - 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 4.44iT - 97T^{2} \) |
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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
−10.36644158055459173611890394190, −10.24650933340457964363552455128, −9.293871947206007969760416534551, −8.321133509363248593547069638526, −7.05275006833341500885039349252, −5.33195578376663913744324793332, −4.49570962617696038617552861540, −3.49703510890838847223918196093, −2.54771212301762541711046823929, −0.77630648674142647658818347821,
1.80531395515927129298793889554, 4.01258524216955506088790076057, 5.00210981434479267325767906747, 6.30599571365374834331701642919, 6.63713441751894087032892949133, 7.59508916265941725592877222876, 8.382554217532235293253610767029, 9.169430507057608980654995886158, 10.21233640645040518357040816073, 11.62737194851000376682928984104
