| L(s) = 1 | + 1.41i·2-s + 2.82i·3-s − 2.00·4-s − 5-s − 4.00·6-s + 5·7-s − 2.82i·8-s + 0.999·9-s − 1.41i·10-s + 5·11-s − 5.65i·12-s − 16.9i·13-s + 7.07i·14-s − 2.82i·15-s + 4.00·16-s − 25·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.942i·3-s − 0.500·4-s − 0.200·5-s − 0.666·6-s + 0.714·7-s − 0.353i·8-s + 0.111·9-s − 0.141i·10-s + 0.454·11-s − 0.471i·12-s − 1.30i·13-s + 0.505i·14-s − 0.188i·15-s + 0.250·16-s − 1.47·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.735362 + 0.735362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735362 + 0.735362i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 2.82iT - 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 7 | \( 1 - 5T + 49T^{2} \) |
| 11 | \( 1 - 5T + 121T^{2} \) |
| 13 | \( 1 + 16.9iT - 169T^{2} \) |
| 17 | \( 1 + 25T + 289T^{2} \) |
| 23 | \( 1 + 10T + 529T^{2} \) |
| 29 | \( 1 - 42.4iT - 841T^{2} \) |
| 31 | \( 1 + 42.4iT - 961T^{2} \) |
| 37 | \( 1 + 25.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 130T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 16.9iT - 9.40e3T^{2} \) |
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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
−16.04461245024572773910857014457, −15.41560992737907837824552420985, −14.43117277388627112367016167710, −13.04614192462643548758131361985, −11.35388525350450538182918253350, −10.08113353934198064649439268782, −8.785817704554103179688929946246, −7.38922964383083705987277679191, −5.44173856762884846101821411921, −4.05897326619392014022796515472,
1.80246226895369231409882706105, 4.39427648108079931839440110889, 6.66675494854769332319275468455, 8.110799446618265920768383417371, 9.568098904237598978724782117449, 11.35769899751648216775133019409, 11.98348497932857562733541939298, 13.39084494492758226668165550448, 14.16627309687701457421252395232, 15.77567872196033378069628506493
