| L(s) = 1 | − 8.04i·5-s − 2·7-s − 21.7i·11-s − 17.3·13-s − 11.8i·17-s − 10.7·19-s + 35.0i·23-s − 39.7·25-s + 11.7i·29-s + 35.5·31-s + 16.0i·35-s − 26·37-s + 2.28i·41-s + 65.5·43-s + 27.2i·47-s + ⋯ |
| L(s) = 1 | − 1.60i·5-s − 0.285·7-s − 1.97i·11-s − 1.33·13-s − 0.697i·17-s − 0.566·19-s + 1.52i·23-s − 1.59·25-s + 0.405i·29-s + 1.14·31-s + 0.459i·35-s − 0.702·37-s + 0.0558i·41-s + 1.52·43-s + 0.578i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6316783997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6316783997\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.04iT - 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 21.7iT - 121T^{2} \) |
| 13 | \( 1 + 17.3T + 169T^{2} \) |
| 17 | \( 1 + 11.8iT - 289T^{2} \) |
| 19 | \( 1 + 10.7T + 361T^{2} \) |
| 23 | \( 1 - 35.0iT - 529T^{2} \) |
| 29 | \( 1 - 11.7iT - 841T^{2} \) |
| 31 | \( 1 - 35.5T + 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 2.28iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 27.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 7.52T + 3.72e3T^{2} \) |
| 67 | \( 1 - 65.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 84.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 75.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 48.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 106.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.224013787541784085975892872510, −8.322173217065292399771156464907, −7.73341857037799949052006921446, −6.47952713502294414229071366743, −5.48410867334203188656611632531, −4.97917310888126054859605780486, −3.85707025730444270589982389739, −2.73857402555764682105509436760, −1.18364812921292370210566993123, −0.19668035345980622432827921601,
2.17144845055516357497557692129, 2.64336087446471826629011314015, 4.02154432303256850825136946439, 4.80881362519043683380618168546, 6.18753681458099974202219078113, 6.91427859028503265299465165544, 7.31105777457403509846354592292, 8.332960929465319961094168530577, 9.645100184726806044303551906079, 10.15114348939119461898127038742
