| L(s) = 1 | − 3.51i·2-s + 0.751i·3-s − 8.35·4-s + 2.64·6-s + 15.3i·8-s + 8.43·9-s + 17.4·11-s − 6.27i·12-s − 25.8i·13-s + 20.4·16-s − 29.6i·18-s − 19·19-s − 61.2i·22-s − 11.5·24-s − 90.9·26-s + 13.0i·27-s + ⋯ |
| L(s) = 1 | − 1.75i·2-s + 0.250i·3-s − 2.08·4-s + 0.440·6-s + 1.91i·8-s + 0.937·9-s + 1.58·11-s − 0.523i·12-s − 1.99i·13-s + 1.27·16-s − 1.64i·18-s − 19-s − 2.78i·22-s − 0.479·24-s − 3.49·26-s + 0.485i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.547759343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547759343\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 3.51iT - 4T^{2} \) |
| 3 | \( 1 - 0.751iT - 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 17.4T + 121T^{2} \) |
| 13 | \( 1 + 25.8iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 44.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 124. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 172. iT - 9.40e3T^{2} \) |
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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
−10.47243660464327583684924697069, −9.756137968811730292105680213493, −9.007843880188140287390518604690, −7.973880939440516764008087824326, −6.54669420226220007536770262130, −5.11935662297974012655820522874, −4.04966302104029589030511121612, −3.35931762688713705653133103932, −1.92374256901489733136857173545, −0.70514932863947419177376555692,
1.55630713772291569045248424094, 4.13665667216125250456537639551, 4.51140421926869257214695972607, 6.12414609653705380268107205214, 6.71137249906567058404331810848, 7.21374197672347818646344007010, 8.458162768126663706333871166401, 9.132606564323301431133304736402, 9.846016893039924728920864278435, 11.38049867826996420083069419021
