| L(s) = 1 | + 1.96i·2-s − 29.7i·3-s + 28.1·4-s + 58.4·6-s − 176. i·7-s + 118. i·8-s − 643.·9-s − 62.7·11-s − 837. i·12-s + 169i·13-s + 347.·14-s + 668.·16-s + 1.82e3i·17-s − 1.26e3i·18-s − 2.50e3·19-s + ⋯ |
| L(s) = 1 | + 0.347i·2-s − 1.90i·3-s + 0.879·4-s + 0.663·6-s − 1.36i·7-s + 0.652i·8-s − 2.64·9-s − 0.156·11-s − 1.67i·12-s + 0.277i·13-s + 0.474·14-s + 0.652·16-s + 1.53i·17-s − 0.919i·18-s − 1.59·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4933304554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4933304554\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 - 1.96iT - 32T^{2} \) |
| 3 | \( 1 + 29.7iT - 243T^{2} \) |
| 7 | \( 1 + 176. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 62.7T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.82e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 137. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.27e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.67e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.94e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.36e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 790. iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.45e3iT - 8.58e9T^{2} \) |
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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
−10.64686780396259115703354517394, −8.691649821817345043721489945482, −7.81442094704853059464585051372, −7.22484367495488484515803069028, −6.49930741654576706668309943845, −5.75701587428554165323098806944, −3.76535894237427656116041944346, −2.18357623263459783557937227795, −1.48730986556751410052349299993, −0.11150717369483002640838670940,
2.36525825410572770755879608990, 3.05540744040590152329196563285, 4.30932565319622453636770288905, 5.42844610650417452514001042372, 6.14988792432795906345773376101, 7.83442040614640277475029111067, 9.157216984587382052070805371534, 9.431029792333897162473216190423, 10.68516924903941286358940554087, 11.11541607837833626055802235058
