| L(s) = 1 | − 6.07i·2-s + 25.6i·3-s − 4.94·4-s + 155.·6-s − 137. i·7-s − 164. i·8-s − 413.·9-s + 169.·11-s − 126. i·12-s + 169i·13-s − 834.·14-s − 1.15e3·16-s + 487. i·17-s + 2.51e3i·18-s − 2.38e3·19-s + ⋯ |
| L(s) = 1 | − 1.07i·2-s + 1.64i·3-s − 0.154·4-s + 1.76·6-s − 1.05i·7-s − 0.908i·8-s − 1.70·9-s + 0.421·11-s − 0.254i·12-s + 0.277i·13-s − 1.13·14-s − 1.13·16-s + 0.408i·17-s + 1.82i·18-s − 1.51·19-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
| \(L(3)\) |
\(\approx\) |
\(0.8343181178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8343181178\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 6.07iT - 32T^{2} \) |
| 3 | \( 1 - 25.6iT - 243T^{2} \) |
| 7 | \( 1 + 137. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 169.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 487. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.38e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.14e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.73e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.91e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.64e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.48e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.27e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.64e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.21e5iT - 8.58e9T^{2} \) |
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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
−10.87874385118669052686812728114, −10.33886649888557775465129395205, −9.621421253475044928351954759514, −8.777312801857995170595357600103, −7.26961981637941221000800025739, −6.00034906093732428536283742014, −4.38928529926288370143696484975, −4.01848971009343847533116733737, −2.99144136971711744263983233692, −1.41823243721137462808196812323,
0.20320156896442999409901519302, 1.87515480593145921068027818920, 2.66653389151328845499909775463, 4.91778359903567618563532810606, 6.18170812831406850587190191681, 6.45108318449258411226025877962, 7.42874489963799323343533642504, 8.439210623721892381742409479095, 8.776387954484754840817112145412, 10.63629028761202070187271341357
