| L(s) = 1 | + (−1.46 + 1.36i)2-s + (3.58 − 0.540i)3-s + (0.298 − 3.98i)4-s + (−5.75 + 14.6i)5-s + (−4.52 + 5.67i)6-s + (−16.0 − 9.21i)7-s + (4.98 + 6.25i)8-s + (−13.2 + 4.07i)9-s + (−11.5 − 29.3i)10-s + (12.1 + 3.75i)11-s + (−1.08 − 14.4i)12-s + (14.6 + 64.3i)13-s + (36.0 − 8.35i)14-s + (−12.7 + 55.7i)15-s + (−15.8 − 2.38i)16-s + (−95.5 + 65.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.480i)2-s + (0.690 − 0.104i)3-s + (0.0373 − 0.498i)4-s + (−0.514 + 1.31i)5-s + (−0.307 + 0.386i)6-s + (−0.867 − 0.497i)7-s + (0.220 + 0.276i)8-s + (−0.489 + 0.151i)9-s + (−0.364 − 0.927i)10-s + (0.333 + 0.102i)11-s + (−0.0260 − 0.348i)12-s + (0.313 + 1.37i)13-s + (0.688 − 0.159i)14-s + (−0.219 + 0.959i)15-s + (−0.247 − 0.0372i)16-s + (−1.36 + 0.929i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.147748 + 0.728793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147748 + 0.728793i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 7 | \( 1 + (16.0 + 9.21i)T \) |
| good | 3 | \( 1 + (-3.58 + 0.540i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (5.75 - 14.6i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (-12.1 - 3.75i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-14.6 - 64.3i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (95.5 - 65.1i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (68.4 + 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-37.9 - 25.8i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-165. - 79.8i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (13.3 - 23.1i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-11.7 - 157. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-252. - 316. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-152. + 191. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-252. + 234. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-50.0 + 668. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-0.386 - 0.985i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (20.4 + 272. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (440. - 762. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-191. + 92.0i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-775. - 719. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (11.0 + 19.1i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (161. - 709. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (358. - 110. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 + 831.T + 9.12e5T^{2} \) |
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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
−14.09343044825159305413184108995, −13.22235321650277472032843963732, −11.36176069260856734452123267749, −10.71052598907077306150604443681, −9.304781845046304255299154586763, −8.431038484911574720684627737046, −6.84656376894886263519608202635, −6.64852543005167336177550872246, −4.05151852891629977952853052962, −2.55296529913755347362992157464,
0.44502532147707725858593727133, 2.72311780760946262504968487951, 4.13429839130221639503910430838, 5.94428324187493165520725337947, 7.893710228172670659628907123495, 8.781199585665390061673443387299, 9.298880956293963097332971015694, 10.74741920350723831913643872587, 12.16263214347395060380299177471, 12.69012225719283951804842547503
