L(s) = 1 | + (−1.5 − 2.59i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 9·5-s + (1.5 − 2.59i)6-s + (−7.5 + 12.9i)7-s − 21·8-s + (13 − 22.5i)9-s + (13.5 + 23.3i)10-s + (24 + 41.5i)11-s − 1.00·12-s + 45·14-s + (−4.5 − 7.79i)15-s + (35.5 + 61.4i)16-s + (−22.5 + 38.9i)17-s − 78·18-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.918i)2-s + (0.0962 + 0.166i)3-s + (−0.0625 + 0.108i)4-s − 0.804·5-s + (0.102 − 0.176i)6-s + (−0.404 + 0.701i)7-s − 0.928·8-s + (0.481 − 0.833i)9-s + (0.426 + 0.739i)10-s + (0.657 + 1.13i)11-s − 0.0240·12-s + 0.859·14-s + (−0.0774 − 0.134i)15-s + (0.554 + 0.960i)16-s + (−0.321 + 0.555i)17-s − 1.02·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.790100 + 0.206286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.790100 + 0.206286i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + 9T + 125T^{2} \) |
| 7 | \( 1 + (7.5 - 12.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-24 - 41.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (22.5 - 38.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-81 - 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-72 - 124. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 264T + 2.97e4T^{2} \) |
| 37 | \( 1 + (151.5 + 262. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-96 - 166. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (48.5 - 84.0i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 111T + 1.03e5T^{2} \) |
| 53 | \( 1 + 414T + 1.48e5T^{2} \) |
| 59 | \( 1 + (261 - 452. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (188 - 325. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-18 - 31.1i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (178.5 - 309. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 830T + 4.93e5T^{2} \) |
| 83 | \( 1 + 438T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-219 - 379. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-426 + 737. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.13248071771579872580522132964, −11.54826496835479020711723946856, −10.30346187888908419046565351261, −9.456198074146389438820582188386, −8.787242315631982601155538409272, −7.22823142167726694641313908047, −6.08866618980640591963573144870, −4.26155161376098057703570560218, −3.05774053755921450511831765440, −1.41027907882468251670775195145,
0.47705480509723496677370257319, 3.08656526809905597924963636554, 4.50659932249002245930000288101, 6.30155493213458042675539235695, 7.09115525756639724753531109780, 8.038687337217344090768644001416, 8.744426198168635485537241731664, 10.10514452258259534157879273019, 11.26139129667520852892350835824, 12.16693277848303983248110182776