L(s) = 1 | − 23.5i·2-s + (−17.0 + 79.1i)3-s − 299.·4-s − 958. i·5-s + (1.86e3 + 401. i)6-s + 3.89e3·7-s + 1.03e3i·8-s + (−5.98e3 − 2.69e3i)9-s − 2.26e4·10-s + 4.41e3i·11-s + (5.10e3 − 2.37e4i)12-s − 4.31e4·13-s − 9.17e4i·14-s + (7.59e4 + 1.63e4i)15-s − 5.23e4·16-s − 3.45e4i·17-s + ⋯ |
L(s) = 1 | − 1.47i·2-s + (−0.210 + 0.977i)3-s − 1.17·4-s − 1.53i·5-s + (1.44 + 0.309i)6-s + 1.62·7-s + 0.253i·8-s + (−0.911 − 0.411i)9-s − 2.26·10-s + 0.301i·11-s + (0.246 − 1.14i)12-s − 1.51·13-s − 2.38i·14-s + (1.49 + 0.322i)15-s − 0.798·16-s − 0.413i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.131884 + 1.24058i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.131884 + 1.24058i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (17.0 - 79.1i)T \) |
| 11 | \( 1 - 4.41e3iT \) |
good | 2 | \( 1 + 23.5iT - 256T^{2} \) |
| 5 | \( 1 + 958. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 3.89e3T + 5.76e6T^{2} \) |
| 13 | \( 1 + 4.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 3.45e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.30e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.06e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.96e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.29e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.31e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.91e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.29e5T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.70e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.81e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 9.40e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.24e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.43e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.67e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.32e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.94e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.72e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.40e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.17e8T + 7.83e15T^{2} \) |
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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
−14.11435065539420520415907344773, −12.42292039071173740268569424399, −11.81514582102128763476467339148, −10.59148092492279730477383221777, −9.443479069399079232184633399219, −8.345048284096258213550855401076, −4.87364275864452620533110475896, −4.48638098238551692416796245480, −2.13216973167682959189216969681, −0.52124123580684148844519540898,
2.22624037412637619459554280271, 5.13572758590351894386506758852, 6.54873267246613103880614945062, 7.43417156429977417124492200197, 8.296289184910174795181148717301, 10.74660810972441174772565235967, 11.83120610588447545821095654383, 13.82049470410866282417314614450, 14.57459224908687094965911063200, 15.12197966113377550949059980758