| L(s) = 1 | + (−4.27 + 0.274i)2-s + (−1.17 − 1.68i)3-s + (10.2 − 1.32i)4-s + (4.43 + 2.88i)5-s + (5.49 + 6.89i)6-s + (4.20 + 0.270i)7-s + (−9.99 + 1.94i)8-s + (7.86 − 21.3i)9-s + (−19.7 − 11.1i)10-s + (−0.707 − 2.38i)11-s + (−14.3 − 15.7i)12-s + (1.85 + 57.9i)13-s − 18.0·14-s + (−0.348 − 10.8i)15-s + (−38.0 + 9.98i)16-s + (−41.5 + 40.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 + 0.0971i)2-s + (−0.226 − 0.324i)3-s + (1.28 − 0.165i)4-s + (0.396 + 0.258i)5-s + (0.373 + 0.468i)6-s + (0.227 + 0.0145i)7-s + (−0.441 + 0.0860i)8-s + (0.291 − 0.791i)9-s + (−0.625 − 0.352i)10-s + (−0.0194 − 0.0653i)11-s + (−0.345 − 0.379i)12-s + (0.0396 + 1.23i)13-s − 0.345·14-s + (−0.00600 − 0.187i)15-s + (−0.594 + 0.155i)16-s + (−0.592 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.802365 - 0.0908597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802365 - 0.0908597i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-817. - 2.64e3i)T \) |
| good | 2 | \( 1 + (4.27 - 0.274i)T + (7.93 - 1.02i)T^{2} \) |
| 3 | \( 1 + (1.17 + 1.68i)T + (-9.32 + 25.3i)T^{2} \) |
| 5 | \( 1 + (-4.43 - 2.88i)T + (50.5 + 114. i)T^{2} \) |
| 7 | \( 1 + (-4.20 - 0.270i)T + (340. + 43.8i)T^{2} \) |
| 11 | \( 1 + (0.707 + 2.38i)T + (-1.11e3 + 726. i)T^{2} \) |
| 13 | \( 1 + (-1.85 - 57.9i)T + (-2.19e3 + 140. i)T^{2} \) |
| 17 | \( 1 + (41.5 - 40.2i)T + (157. - 4.91e3i)T^{2} \) |
| 19 | \( 1 + (-12.5 + 6.04i)T + (4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (-22.0 + 136. i)T + (-1.15e4 - 3.83e3i)T^{2} \) |
| 29 | \( 1 + (-161. - 178. i)T + (-2.34e3 + 2.42e4i)T^{2} \) |
| 31 | \( 1 + (-78.5 - 31.8i)T + (2.14e4 + 2.07e4i)T^{2} \) |
| 37 | \( 1 + (-426. - 111. i)T + (4.41e4 + 2.48e4i)T^{2} \) |
| 41 | \( 1 + (-240. + 232. i)T + (2.20e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-83.7 - 282. i)T + (-6.66e4 + 4.33e4i)T^{2} \) |
| 47 | \( 1 + (-229. + 439. i)T + (-5.93e4 - 8.51e4i)T^{2} \) |
| 53 | \( 1 + (-11.9 + 123. i)T + (-1.46e5 - 2.84e4i)T^{2} \) |
| 59 | \( 1 + (-433. + 244. i)T + (1.06e5 - 1.75e5i)T^{2} \) |
| 61 | \( 1 + (-206. + 295. i)T + (-7.83e4 - 2.13e5i)T^{2} \) |
| 67 | \( 1 + (391. - 749. i)T + (-1.72e5 - 2.46e5i)T^{2} \) |
| 71 | \( 1 + (-228. + 621. i)T + (-2.72e5 - 2.32e5i)T^{2} \) |
| 73 | \( 1 + (-489. - 128. i)T + (3.38e5 + 1.90e5i)T^{2} \) |
| 79 | \( 1 + (-310. + 202. i)T + (1.99e5 - 4.50e5i)T^{2} \) |
| 83 | \( 1 + (-1.14e3 - 552. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (901. - 364. i)T + (5.06e5 - 4.90e5i)T^{2} \) |
| 97 | \( 1 + (-373. - 616. i)T + (-4.22e5 + 8.09e5i)T^{2} \) |
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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
−11.68535037531631130016237153599, −10.79596774457386616771343218377, −9.863157556435448077290395058598, −9.027185217073459654108432155846, −8.172525483773965715713445406346, −6.81225765851545200443870099674, −6.40605689454880138652231015418, −4.37548039908337190100078721356, −2.21488236949949676131726119787, −0.862602385501125397579068821868,
0.931401746792171683583577768046, 2.44812261250952228040452896458, 4.61832492849827091942251754460, 5.84546450144389826597058371097, 7.49633825190118363652479689957, 8.034389311108303513648236619440, 9.331189084350766824018526110092, 9.908232074619482298580693028540, 10.85585576275673326449153928310, 11.52046738347322849901216004540
